N=4 Supersymmetric Yang–Mills Theory and the Ads/SCFT

Home , Instanton, Supersymmetry

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FACOLTA` DI SCIENZE MATEMATICHE, FISICHE E NATURALI Dipartimento di Fisica

=4 supersymmetric Yang–Mills theory N and the AdS/SCFT correspondence

Tesi di dottorato di ricerca in Fisica presentata da Stefano Kovacs

Relatori Professor Giancarlo Rossi Professor Massimo Bianchi

Coordinatore del dottorato Professor Piergiorgio Picozza

Ciclo XI

Anno Accademico 1997-1998 Contents

Acknowledgments v

Note added v

1 Rigid supersymmetric theories in four dimensions 1 1.1 General supersymmetry algebra in four dimensions ...... 2 1.2 Representations of the supersymmetry algebra ...... 5 1.3 =1superspace ...... 10 N 1.4 =1supersymmetricmodels ...... 17 N 1.4.1 Supersymmetric chiral theories: Wess–Zumino model . .... 18 1.4.2 Supersymmetricgaugetheories ...... 19 1.5 Theories of extended supersymmetry ...... 22 1.5.1 Extended superspace: a brief survey ...... 23 1.5.2 > 1supersymmetricmodels...... 27 N 1.6 Quantization of supersymmetric theories ...... 29 1.6.1 Generalformalism ...... 30 1.6.2 Superﬁeldpropagators ...... 33 1.6.3 Feynmanrules...... 35 1.7 General properties of supersymmetric theories ...... 36

2 =4 supersymmetric Yang–Mills theory 40 N 2.1 Diverse formulations of =4 supersymmetric Yang–Mills theory . . . 41 N 2.2 Classical symmetries and conserved currents ...... 46 2.3 Finiteness of =4 supersymmetric Yang–Mills theory ...... 50 N 2.4 S-duality...... 52 2.4.1 Montonen–Olive and SL(2,Z)duality ...... 54 2.4.2 S-duality in =4 supersymmetric Yang–Mills theory . . . . . 57 N ii Table of contents iii

2.4.3 Semiclassical quantization ...... 60

3 Perturbative analysis of =4 supersymmetric Yang–Mills theory 66 N 3.1 Perturbation theory in components: problems with the Wess–Zumino gauge...... 68 3.1.1 One loop corrections to the propagator of the fermions belong- ingtothechiralmultiplet ...... 72 3.1.2 One loop corrections to the propagator of the scalars belonging tothechiralmultiplet ...... 74 3.2 Perturbation theory in =1superspace: propagators ...... 79 N 3.2.1 Propagatorofthechiralsuperfield...... 81 3.2.2 Propagatorofthevectorsuperfield ...... 86 3.2.3 Discussion...... 93 3.3 Three- and four-point functions of =1superfields ...... 96 N 3.3.1 Three-pointfunctions...... 96 3.3.2 Four-pointfunctions ...... 99

4 Nonperturbative eﬀects in =4 super Yang–Mills theory 108 N 4.1 Instanton calculus in non Abelian gauge theories ...... 110 4.1.1 Yang–Millsinstantons ...... 111 4.1.2 Semiclassical quantization ...... 114 4.1.3 Bosoniczero-modes...... 116 4.2 Instanton calculus in the presence of fermionicﬁelds ...... 118 4.2.1 Semiclassical approximation ...... 119 4.2.2 Fermioniczero-modes...... 121 4.3 Instanton calculus in supersymmetric gaugetheories...... 123 4.4 Instanton calculations in =4 supersymmetric Yang–Mills theory . . 125 N 4.5 A correlation function of four scalar supercurrents ...... 126 4.6 Eight- and sixteen-point correlation functions of current bilinears . . 136 4.6.1 The correlation function of sixteen fermionic currents ..... 136 4.6.2 The correlation function of eight gaugino bilinears ...... 137 4.7 Generalization to arbitrary N and to any K for large N ...... 138 iv Table of contents

5 AdS/SCFT correspondence 141 5.1 Type IIB superstring theory: a bird’s eye view ...... 142 5.2 D-branes...... 147 5.3 The AdS/SCFT correspondence conjecture ...... 151 5.4 Calculations of two- and three-point functions ...... 160 5.5 Four-pointfunctions ...... 170 5.6 D-instanton eﬀects in d=10 type IIB superstring theory ...... 177 5.6.1 Testing the AdS/CFT correspondence: type IIB D-instantons vs. Yang–Millsinstantons ...... 181 5.7 The D-instanton solution in AdS S5 ...... 184 5 × Conclusions 189

A Notations and spinor algebra 193

B Electric-magnetic duality 198 B.1 Electric-magnetic duality and the Dirac monopole ...... 198 B.2 Non Abelian case: the ’t Hooft–Polyakov monopole ...... 200 B.3 Monopolesandfermions ...... 204

Bibliography 207 Acknowledgments

This dissertation is based on research done at the Department of Physics of the Uni- versit`adi Roma “Tor Vergata”, during the period November 1995 - October 1998, under the supervision of Prof. Massimo Bianchi and Prof. Giancarlo Rossi. I would like to thank them for introducing me to the arguments covered in this thesis, for all what they have taught me and for their willingness and constant encouragement. I also wish to thank Prof. Michael B. Green for the very enjoyable and fruitful collaboration on the subjects reported in the last two chapters. I am grateful to Yassen Stanev for many useful and interesting discussions. Finally I thank the whole theoretical physics group at the Department of Physics of the Universit`adi Roma “Tor Vergata” for the very stimulating environment I have found there.

Note added

This thesis was presented to the “Universitàdi Roma Tor Vergata” in candidacy for the degree of “Dottore di Ricerca in Fisica” and it was defended in April 1999. Hence more recent developments are not discussed here and moreover the references are not updated. However I cannot help mentioning at least the very detailed and comprehensive review by O. Arhony, S. Gubser, J. Maldacena, H. Ooguri and Y. Oz on “Large N Field Theories, String Theory and Gravity” (hep-th/9905111), that is not cited in the bibliography. Actually this paper would have made my writing of the last chapter much simpler if it came out earlier! There are some direct developments of this thesis that have been carried on during the last few months. In particular the analysis of logarithmic singularities in four- point Green functions and their relation to anomalous dimensions of un-protected operators in =4 SYM has been finalised in collaboration with Massimo Bianchi, N Giancarlo Rossi and Yassen Stanev. This issue is only briefly addressed in the concluding remarks. The interested reader is referred to the paper “On the logarithmic behaviour in =4 SYM theory”, hep-th/9906188. N

v Chapter 1

Rigid supersymmetric theories in four dimensions

Supersymmetry was introduced in [1] as a generalization of Poincaréinvariance and then implemented in the context of four dimensional quantum field theory in [2]. The idea of a symmetry transformation exchanging bosonic and fermionic degrees of freedom has proved extremely rich of consequences. Although up to now there exists no experimental evidence for supersymmetry, it plays a central rôle in theoretical physics. At the present day the interest in the study of supersymmetric theories relies on one side on the rôle it plays in the construction of a more fundamental model beyond the standard model of particle physics (SM) and on the other on the possibility of deriving results that can hopefully be extended to phenomenologically more relevant models. The standard model describes the strong and electro-weak interactions within the framework of a SU(3) SU(2) U(1) non-Abelian gauge theory; despite the full × × agreement with experimental data there are various motivations for believing that it cannot be a truly fundamental theory. A first problem arises in relation with the grand unification of particle interactions. The gauge couplings are believed to be unified at an energy scale of the order of 1016 GeV. The standard model contains many free parameters (masses and mixing angles) that must be “fine-tuned” in order to achieve the grand unification (hierarchy problem): the problem follows from the

1 2 Chapter 1. Rigid 4d supersymmetry

existence of ultra-violet quadratic divergences and is solved in supersymmetric models in which the divergences, as will be discussed later, are milder. Moreover the standard model does not include gravitational interactions: at present the only consistent description of gravity at the quantum level is provided by (super)string theories and supersymmetry appears to be a fundamental ingredient of such theories. In a more immediate perspective supersymmetric theories can be of help in understanding aspects of phenomenologically interesting models. For example a description of confinement in asymptotically free theories as dual superconductivity has been proposed and various realizations of this mechanism have been observed in supersymmetric models. More recently a correspondence between four dimensional quantum field theories and supergravity has been suggested; this particular subject appears very promising and will be extensively studied in chapter 5. This first chapter contains a review of rigid supersymmetry in four dimensions. The material presented is by no means original, the aim is to establish the notations and to recall ideas and results, mainly about the quantization of supersymmetric theories, that will be extensively used in the following. In the first part of the chapter the supersymmetry algebra and its realizations are described. Then the formalism of superspace is introduced in somewhat more detail both at the classical and at the quantum level. The last section reviews general properties of supersymmetric models. An introduction to supersymmetry is provided by the books by P. West [3] and J. Wess and J. Bagger [4] (whose notations will be followed in this thesis) and by the review papers [5, 6]; superspace techniques are treated in more detail in [7]; for phenomenological aspects of supersymmetry see [8]; an extensive introduction to superstring theory is given in the book by M.B. Green, J.H. Schwarz and E. Witten [9].

1.1 General supersymmetry algebra in four dimensions

Quantum ﬁeld theory models of interest for the description of fundamental interactions combine the Poincar´einvariance with the invariance under a global internal symmetry group G. In [10] Coleman and Mandula proved, under general assumptions 1.1 Supersymmetry algebra 3

following from the axioms of quantum ﬁeld theory, a “no-go” theorem that, states that the most general symmetry group for a theory with non trivial S-matrix is the direct product of the Poincar´egroup and an internal group G, which must be of the form of a semisimple Lie group times possible U(1) factors. Historically supersymmetry was discovered as an attempt to evade the constraints of the Coleman–Mandula no-go theorem .

The Poincar´egroup P=ISO(3,1) is the semidirect product of the group T4 of four dimensional translations and the Lorentz group SO(3,1); it is generated by Pµ, Mµν satisfying:

[Pµ,Pν]=0 [P , M ]=(η P η P ) (1.1) λ µν λµ ν − λν µ [M , M ]= (η M + η M η M η M ) . µν ρσ − µρ νσ νσ µρ − µσ νρ − νρ µσ

(See appendix A for the conventions and a collection of useful formulas). The generic internal symmetry group is a Lie group deﬁned by the commutation relations:

a b ab c [T , T ]= if cT . (1.2)

Internal symmetries are supplemented by the discrete ones C, P and T. The supersymmetry algebra is a graded extension of (1.1): the theorem of Cole- man and Mandula is bypassed by allowing anticommuting as well as commuting generators. In [11] the most general algebra allowed by this weakening of the hypothesis of [10] was constructed: it contains, beyond (1.1) and (1.2), anticommuting i 1 1 generators Qα, Qαj˙ that are Weyl spinors transforming in the ( 2 , 0) and (0, 2 ) representations of the Lorentz group respectively, so that they do not commute with the generators Pµ and Mµν . Here α, α˙ =1, 2 are spinor indices and i, j ranging from 1 to 1 label the various of supersymmetries. N ≥ The most general symmetry algebra of a supersymmetric quantum ﬁeld theory is therefore given by (1.1) and (1.2) supplemented by

j Qi , Q =2σµ P δij { α β} αα˙ µ i [Pµ, Qα] = [Pµ, Qαj˙ ]=0 i β i [Qα, Mµν]=(σµν )α Qβ 4 Chapter 1. Rigid 4d supersymmetry

˙ α˙ β [Qαj˙ , Mµν ]=(σµν ) β˙ Qj Qi , Qj = ε Zij { α β} αβ Q , Q ˙ = ε ˙ Z† (1.3) { αi˙ βj} α˙ β ij i a ai j [Qα, T ]= B jQα a ai [T , Qαi˙ ]= Bj† Qαi˙ ij [Z ,X] = [Zlk† ,X]=0 ,

aj aj where Bi† = B i. In (1.3) X denotes an arbitrary generator in the algebra so that Zij = Zji (central charges) generate the center of the algebra 1. Central charges − play a major rˆole in the quantization of -extended supersymmetric theories. They N generate an Abelian subalgebra so that

ij ij b Z = ab T ,

with Bi abjk = ab ijB k. aj − aj The supersymmetry algebra (1.3) considered as a graded Lie algebra possesses a group of automorphisms which is U( ) in the general case . The U( ) group of N N i automorphisms acts on the charges Qα and Qαj˙ as

i i k k Q U Q Q Q U † . α −→ k α αj˙ −→ αk˙ j In the particular case of =1 supersymmetry the U(1) group of automorphisms is N generated by R such that

[Q , R]= Q , [Q , R]= Q . α α α˙ − α˙

i 1 The supersymmetry charges Qα and Qαj˙ transform in spin 2 representations of the i Lorentz group; this implies that acting with Qα or Qαj˙ on a state of spin j produces a state of spin j 1 : supersymmetry generators exchange bosonic and fermionic ± 2 states. Reversing the line of reasoning and starting with a symmetry transforming the fields among themselves in such a way as to mix bosons and fermions, one concludes that the different physical dimension of bosonic and fermionic fields implies that the transformation must involve derivatives, i.e. space-time translations. This gives an 1In principle the antisymmetry of Zij is expected to imply that no central extension can exist in non extended ( =1) theories. It can be proved [12] that a central extension can actually be present in =1 supersymmetricN Yang–Mills theory. See [13] for a recent discussion in the context of the AdS/CFTN correspondence. 1.2 Representations of supersymmetry 5

intuitive explanation of how the requirement of a non trivial mixing of internal and space-time symmetries naturally leads to supersymmetry. An immediate consequence of the supersymmetry algebra is that in supersymmetric theories the energy is positive deﬁnite and vanishes only on supersymmetric (ground) states. The ﬁrst equation in (1.3) implies

i i 0 Q , (Q )† + Q , (Q )† = 4 P =4 H , ≤ { 1 1i } { 2 2i } − N 0 N i X so that

ψ H ψ 0 h | | i ≥ for every state ψ and H Ψ = 0 if and only if Q ψ = 0, i.e. if ψ is supersymmetric. | i | i | i | i 1.2 Representations of the supersymmetry algebra

The irreducible representations of the general supersymmetry algebra can be constructed starting from the (anti)commutation relations of the preceding section and the definition of the Casimir operators. For the Poincarégroup the quadratic Casimir 2 µ 2 µ operators are P = PµP and W = WµW , where Wµ is the Pauli–Lubanski vector 1 W = ε P νM ρσ . µ 2 µνρσ 2 Here W is not a Casimir anymore, because Mµν does not commute with the su- i 2 persymmetry generators Qα and Qαj˙ , and is substituted by C defined (for the case =1) by N 2 µν C = Cµν C C = B P B P µν µ ν − ν µ 1 B = W Q σαα˙ Q . µ µ − 4 α˙ µ α The irreducible representations can be constructed by Wigner’s technique of induced representations. A general result following from the supersymmetry algebra is that every irreducible representation contains an equal number of bosonic and fermionic states. Defining the fermion number operator ( )Nf such that − ( )Nf B =+ B , ( )Nf F = F , − | i | i − | i −| i 6 Chapter 1. Rigid 4d supersymmetry

where B denotes a bosonic state and F a fermionic one, it follows that | i | i ( )Nf Q = Q ( )Nf , ( )Nf Q = Q ( )Nf . − α − α − − α˙ − α˙ − These relations imply

0=tr ( )Nf Q , Q =2σµ δijP tr( )Nf , − { α α˙ } αα˙ µ − so that in conclusion for non vanishing Pµ

tr( )Nf =0 − from which the statement follows.

Massive representations without central charges.

For particles of mass M in the rest frame P = ( M, 0, 0, 0) the supersymmetry µ − algebra reduces to

i i Q , Q ˙ =2Mδ δ { α βj} αα˙ j Q , Q = Q Q ˙ =0 , (1.4) { α β} { αi˙ βj} and creation and annihilation operators can be deﬁned as follows 1 ai = Qi α √2M α 1 ai † = Q . (1.5) α √2M αi˙ i i aα and (aα)† satisfy the algebra of fermionic creation and annihilation operators. In fact by direct calculation one ﬁnds

ai , aj † = δβδi { α β } α j j j ai , a = ai † , a † =0 . (1.6) { α β} { α β } Given a Cliﬀord vacuum Ω deﬁned by | i ai Ω =0 i, α , α| i ∀ with P 2 Ω = M 2 Ω , the states are constructed by acting with the operators (ai )† | i − | i α on Ω : | i (n) α ...α 1 Ω 1 n = ain † . . . ai1 † Ω . (1.7) | i1...in i √n! αn α1 | i 1.2 Representations of supersymmetry 7

Antisymmetry under the exchange of pairs (αk, ik), (αl, il) implies n 2 . For each 2 ≤ N n the number of states (degeneracy) is , so that the total number of states nN (i.e. the dimension of the representation) is

2 N 2 2 d = N =2 N . n n=0 X 2 1 2 1 The representation contains 2 N− bosonic and 2 N− fermionic states. The highest spin state is obtained symmetrizing the maximum number of spinor indices, namely , leading to the value 1 for the highest spin in the multiplet. N 2 N

Massless representations without central charges.

In the “light-like” reference frame P =( E, 0, 0, E), with P 2 = 0, the supersymme- µ − try algebra reads

i 2E 0 i Q , Q ˙ = 2 δ { α βj} 0 0 j i j Q , Q = Q , Q ˙ =0 . { α β} { αi˙ βj} The natural deﬁnition of creation and annihilation operators is then 1 ai = Qi 2√E 1 i 1 i a† = (a )† = Q˙ . (1.8) i 2√E 1 They generate the Cliﬀord algebra

i i a , a† = δ { j} j i j a , a = a†, a† =0 (1.9) { } { i j} i Notice that in this case only half of the generators Qα can be used to construct the i fermionic oscillators because Q2 and Q2˙j anticommute. Starting with a state of lowest helicity λ, Ω , playing the rôle of a Clifford vacuum, defined by | λi ai Ω =0 | λi the states which constitute the multiplet are constructed as

(n) 1 n † † Ωλ+ ;i ,...,i = ain ...ai Ωλ . (1.10) | 2 1 n i √n! 1 | i 8 Chapter 1. Rigid 4d supersymmetry

(n) n n The state Ωλ+ ;i ,...,in has helicity λ + 2 and because of antisymmetry in the indices | 2 1 i

i1,... ,in is N times degenerate. The highest helicity in the multiplet is λ = n λ + N2 . The dimension of the representation given by the total number of states is

N d = =2N . Nn i=0 X CPT invariance implies in general a doubling of the number of states because it re- verses the helicity. So for example the =3 multiplet in four dimensions coincides N with the =4 multiplet if CPT invariance is required. Multiplets of particular rel- N evance such as =2 with λ = 1 , =4 with λ = 1 and =8 with λ = 2 are N − 2 N − N − automatically CPT invariant.

Supersymmetry representations in the presence of central charges.

For P 2 = M 2 in the rest frame P =( M, 0, 0, 0) the supersymmetry algebra with − µ − non vanishing central charges is given by

Qi , Q = 2Mδi δ { α αj˙ } j αα˙ Qi , Qj = ε Zij (1.11) { α β} αβ Q , Q ˙ = ε ˙ Z† , { αi˙ βj} − α˙ β ij with Zij = Zji, Z = Zij. Since Zij commute with all the generators, by invoking − ij − Schuur’s lemma there exists a basis in which they are diagonal with eigenvalues Zij. The antisymmetric matrix with elements Zij can be put in the following form N×N ij i kl T j Z = U kZ˜ (U )l ,

with U is a unitary matrix and

Z˜ = ε D (for even) ⊗ N ε D 0 Z˜ = ⊗ (for odd) , 0 0 N 0 1 where ε = and D is a diagonal N N matrix with eigenvalues Z , 1 0 2 × 2 R − such that Z∗ = Z , Z > 0. Considering the case with even the algebra can be R R R N rewritten in a more suitable form by decomposing the indices i and j as

i =(a, R) , j =(b, S) , 1.2 Representations of supersymmetry 9

with a, b =1, 2 and R,S =1,..., N2 . Now the algebra can be put in the form

aR bS ab RS Q , Q ˙ = 2Mδ δ δ { α β } αα˙ QaR, QbS = ε εabδRS Z (1.12) { α β } αβ S aR bS ab RS Q , Q ˙ = ε ˙ ε δ Z , { α˙ β } α˙ β S so that creation and annihilation operators can be deﬁned in the following way:

R 1 1R 2R R R a = Q + εαβ(Q )† , (a )† = a† α √2 α β α α R 1 1R 2R R R b = Q εαβ(Q )† , (b )† = b† α √2 α − β α α These operators satisfy the algebra of fermionic creation and annihilation operators

aR, aS = bR, bS = aR, bS =0 { α β } { α β } { α β } R S RS a , (a )† = δ δ (2M + Z ) (1.13) { α β } αβ S R S RS b , (b )† = δ δ (2M Z ) . { α β } αβ − S

Since A, A† is a positive definite operator A, it immediately follows that Z 2M. { } ∀ R ≤ For Z < 2M R the structure of the multiplet is the same as in the case of no central R ∀ charges. For ZR =2M with R =1,..., R a subset of operators b cancels out and the algebra reduces to a 2( R) dimensional Clifford algebra giving rise to so called N − short multiplets. States belonging to short multiplets, also referred to as Bogomol’nyi Prasad Sommerfield (BPS) saturated states, play a crucial rôle in the non perturbative dynamics of supersymmetric gauge theories as will be discussed later. As an example of the previous construction consider the simplest case, namely =1 massive multiplet with a spin zero Clifford vacuum Ω : the states are N | i Ω (spin 0) | i 1 (a )† Ω (spin ) α | i 2 1 1 γ (a )†(a )† Ω = ε (a )†(a )† Ω (spin 0) √2 α β | i − 2√2 αβ γ | i In the following the discussion will focus mainly on the =4 massless case. For N a rigid supersymmetric theory with fields of spin not larger than one the multiplet contains the following states with relative degeneracies 10 Chapter 1. Rigid 4d supersymmetry

helicity 1 1 0 1 1 − − 2 2 degeneracy 1 4 6 4 1

Given the irreducible representations of the supersymmetry algebra the corresponding realizations in terms of fields can be immediately deduced. Although this was the historical pattern in the construction of supersymmetric quantum field theory models, it appears more natural to derive the structure of supermultiplets using the superfield formalism, so this will be the approach followed here. Notice that Wigner’s method here employed gives representations of “on-shell” states. It is always possible to construct sets of fields realizing such irreducible representations, but on the contrary it is usually not possible, for > 1, to achieve the N closure of the supersymmetry algebra on a set of “off-shell” fields, i.e. without use of the equations of motion. The irreducible representations of supersymmetry have automatically a well defined transformation under the group of automorphisms of the algebra (1.3). For massless representations the largest subgroup of the automorphisms group that re- spects helicity is the whole U( ), whereas for massive states it depends on the N presence and number of central charges. In the absence of central charges, or if none of them saturates the BPS bound, the spin preserving automorphisms group is USp(2 ). It reduces to USp( ) for even and to USp( + 1) for odd if one N N N N N central charge satisfies the condition Z =2M.

1.3 =1 superspace N Ordinary quantum field theories are constructed in terms of field operators that are functions (actually distributions) defined on Minkowskian space-time, parametrized

by the coordinates xµ. The action is required to be invariant under Poincar´espace- time symmetry as well as under transformations of an internal group G. The transformation of a generic ﬁeld f(x) is of the form

R(g)f(x)= f(τ(g)x) , (1.14)

with R(g) and τ(g) suitable representations of the relevant symmetry group. In supersymmetric models the space-time symmetry is extended by the intro- i duction of the transformations generated by anticommuting charges Qα, Qαj˙ . The 1.3 Superspace 11

natural way to implement the extended symmetry algebra is to construct the theory in terms of generalized ﬁelds, called superﬁelds [14, 15], depending on the coordinates xµ as well as on “fermionic” coordinates θα and θα˙

F = F (xµ, θα, θα˙ ) . (1.15)

Starting from the supersymmetry algebra a generic element of the corresponding group of transformations can be written in the form

µ α α˙ g(x, θ, θ)= ei(x Pµ+θ Qα+θα˙ Q ) .

The action on superfields is then defined through a differential representation of the α˙ operators Pµ, Qα and Q . This construction can be carried out in a more general fashion, suitable for generalization to the case of -extended supersymmetry, defining the superspace N parametrized by

zm =(xµ, θα, θα˙ ) as a coset space. =1 superspace under consideration can be obtained as the coset N SP/L, where SP is the super Poincar´egroup generated by (1.3) and L is the Lorentz group SO(3,1). A coset group K=G/H can be parametrized by coordinates ξ , m =1,..., (dim G m − dim H). By writing an element of G through the exponential map

m k g = eiξ Lm eiζ Hk , where the generators of G split into the generators of H (Hk) and the remaining Ln, elements of the coset are obtained by taking ζk = 0. For the present case K=SP/L this implies that one should write

α˙ i(aµP +ηαQ +η Q ) 1 wµν M g = e µ α α˙ e 2 µν (1.16) so that wµν = 0 gives the elements of K

µ α α˙ m k = ei(x Pµ+θ Qα+θα˙ Q ) = ez Km which are parametrized by z =(x , θ , θ ); as a result =1 superspace is an eight m µ α α˙ N dimensional manifold. Under the action of a group element g0 =(aµ, ηα, ηα˙ )

m ′m 1 ′µν z Km z Km w Mµν g e = e e 2 g′ . 0 ◦ ≡ 12 Chapter 1. Rigid 4d supersymmetry

An explicit calculation using Becker–Hausdorﬀ’s formula and anticommutation of Q

and P gives the following transformations of the coordinates zm

µ µ µ α µ α˙ α µ α˙ µν x′ = x + a + iη σ θ iθ σ η + w x αα˙ − αα˙ ν α α α 1 µν α θ′ = θ + η + w σ θ (1.17) 4 µν β β α˙ α˙ 1 β˙ θ′ = θ + ηα˙ θ σµν α˙ w . − 4 β˙ µν

m m Now comparison of (g g ) ez Km with g (g ez Km ) allows one to prove that 1 ◦ 2 ◦ 1 ◦ 2 ◦ the generators Q and Q satisfy the supersymmetry algebra (1.3) for =1. α α˙ N As already mentioned a superﬁeld is a function of zm

F (z)= F (x, θ, θ) .

The transformation under the group action is the generalization of (1.14) that imple-

ments z′ = τ(g)z with z′ given by (1.17). For inﬁnitesimal transformations

δ F = F (z + δz) F (z)= δg XmF (z) , g − m which is achieved realizing the super Poincar´egenerators through the diﬀerential

operators Xm =(ℓm,ℓα, ℓα˙ ,ℓµν )

ℓµ = ∂µ ∂ α˙ ℓ = + iσµ θ ∂ α ∂θα αα˙ µ ∂ α µ ℓα˙ = α˙ + iθ σαα˙ ∂µ (1.18) ∂θ ˙ 1 β µν α ∂ 1 β µνα˙ ∂ ℓµν = (xµ∂ν xν ∂µ) θ σβ + θ σ ˙ α˙ . − − − 2 ∂θα 2 β ∂θ A calculation shows that they satisfy the super Poincar´ealgebra

ℓ ,ℓ =0 , ℓ , ℓ =2iσµ ℓ , { α β} { α α˙ } αα˙ µ (1.19) 1 [ℓ ,ℓ ]=0 , [ℓ ,ℓ ]= σ βℓ . µ α α µν 2 µνα β Superfields defined in this way form linear representations of supersymmetry. Since supersymmetry acts on superfields through linear differential operators it follows that the product of superfields is a new superfield. 1.3 Superspace 13

The objects considered up to now are scalar superfields 2; more general superfields transforming in a non trivial way under Lorentz transformations can be considered as well. In general a superfield carrying a set of Lorentz indices encoded in a single label p transforms as

1 µν q w Mµν R(g)fp(z)= Dp e− 2 Fq(τ(g)z) , q where Dp is a suitable representation of the Lorentz group. Physical fields directly related to the states considered in section 1.2 can be obtained from the superfields by expanding in the θ and θ variables. Since θα and θα˙ are Weyl spinors the expansion reduces to a polynomial. For a generic superfield

α α˙ α α˙ F (x, θ, θ) = f(x)+ θ χα(x)+ θα˙ ψ (x)+ θ θαg(x)+ θα˙ θ h(x)+ α µ α˙ α α˙ α˙ α α α˙ +θ σαα˙ θ rµ(x)+ θ θαθα˙ λ (x)+ θα˙ θ θ ξα(x)+ θ θαθα˙ θ s(x) . (1.20)

The fields f(x), χ(x), ψ(x), g(x), h(x), rµ(x), λ(x), ξ(x) and s(x) constitute a supermultiplet and their transformation law is derived from the transformation of the superfield by defining

α α˙ α α˙ δηF (x, θ, θ)= δηf(x)+ θ δηχα(x)+ θα˙ δηψ (x)+ θ θαδηg(x)+ θα˙ θ δηh(x)+ α µ α˙ µ α α˙ α˙ α α α˙ +θ σαα˙ θ δηr (x)+ θ θαθα˙ δηλ (x)+ θα˙ θ θ δηξα(x)+ θ θαθα˙ θ δηs(x) (1.21) and matching terms with the same powers of θ and θ. Linear representations of supersymmetry constructed in this way are in general reducible; the irreducible representations studied in section 1.2 are in correspondence with constrained superfields. All the representations of rigid =1 supersymmetry in N four dimensions are realized in terms of two basic superfields, namely chiral superfields and vector superfields.

From now on the differential operators ℓα and ℓα˙ will be denoted as Qα and Qα˙ like the abstract generators since they satisfy the same algebra. The operators Qα and Qα˙ realize the left regular representation of supersymmetry on superfields, it is useful to analogously define the right regular representation realized by the differential operators

∂ µ α˙ ∂ α µ Dα = + iσαα˙ θ ∂µ , Dα˙ = α˙ iθ σαα˙ ∂µ (1.22) ∂θα −∂θ − 2From now on scalar superﬁelds will be referred to simply as superﬁelds. 14 Chapter 1. Rigid 4d supersymmetry

satisfying

D , D = 2iσµ ∂ { α α˙ } − αα˙ µ D ,D = D , D ˙ =0 . { α β} { α˙ β}

Dα and Dα˙ are usually referred to as superspace covariant derivatives, actually they can be obtained from the general expression for the covariant derivative in the coset

space SP/L: the fact that Dα and Dα˙ do not reduce to ∂α and ∂α˙ despite the ﬂatness of the =1 superspace is a consequence of the presence of a non vanishing torsion. N

Chiral superﬁelds.

Chiral superﬁelds are characterized by the constraint

Dα˙ Φ(x, θ, θ)=0 ,

which takes a particularly simple form expressing Φ(x, θ, θ), D and D in terms of the variable yµ = xµ + iθσµθ. As functions of y, θ and θ one gets

∂ α˙ ∂ D = +2iσµ θ α ∂θα αα˙ ∂yµ ∂ Dα˙ = α˙ , −∂θ so that the general chiral superﬁeld assumes the form

Φ(y, θ)= φ(y)+ √2θψ(y)+ θθF (y) .

Substituting yµ = xµ + iθσµθ and expanding in θ and θ yields

µ Φ(x, θ, θ) = φ(x)+ √2θψ(x)+ iθσ θ∂µφ(x)+

i µ 1 +θθF (x) θθ∂µψ(x)σ θ + θθθθ2φ(x) . (1.23) − √2 4 Analogously antichiral superﬁelds are deﬁned by

DαΦ†(x, θ, θ)=0 .

µ µ µ µ The natural variable is y† = x iθσ θ; in terms of y† −

Φ† = Φ†(y†, θ)= φ∗(y†)+ √2θψ(y†)+ θθF ∗(y†) . 1.3 Superspace 15

The field components associated to chiral and antichiral multiplets are of two complex scalars (φ, F ) and one Weyl spinor (ψ) corresponding to twice the number of states in the massive on-shell =1 multiplet of section 1.2. N Products of (anti) chiral superfields are still (anti) chiral superfields. For example

ΦiΦj = φi(y)φj(y)+ √2θ[ψi(y)φj(y)+ φi(y)ψj(y)]+ +θθ[φ (y)F (y)+ φ (y)F (y) ψ (y)ψ (y)] i j j i − i j 3 3 The product Φ†Φ is neither chiral nor antichiral. Moreover since D = D = 0 it immediately follows that for a generic superﬁeld F

2 D F = Φ is chiral 2 D F = Φ† is antichiral .

Notice that the double constraint DαΦ= Dα˙ Φ = 0 implies that Φ is constant.

Vector superﬁelds.

Vector superﬁelds obey the constraint

V †(x, θ, θ)= V (x, θ, θ) , which gives rise to the component expansion i V (x, θ, θ)= C(x)+ iθχ(x) iθχ(x)+ θθS(x)+ − √2

i µ i µ θθS†(x) θσ θA (x)+ iθθθ λ(x)+ σ ∂ χ(x) + −√ − µ 2 µ 2 i 1 1 iθθθ λ(x)+ σµ∂ χ(x) + θθθθ D(x)+ 2C(x) . (1.24) − 2 µ 2 2 The supermultiplet contains two real scalars (C, D), one complex scalar (S), four

Weyl spinors (χ, χ, λ, λ) and one real vector (Aµ). This is twice the content of the massive =1 multiplet obtained starting with a Cliﬀord vacuum of spin 1 and also N 2 of the massless CPT invariant multiplet with minimum helicity λ = 1. − Since the vector multiplet contains a vector ﬁeld Aµ(x) it is the basic ingredient for the construction of supersymmetric gauge theories. The supersymmetric generalization of an Abelian gauge transformation [16, 17] is

V V +Φ+Φ† , −→ 16 Chapter 1. Rigid 4d supersymmetry

with

µ Φ + Φ† = φ + φ∗ + √2(θψ + θψ)+ θθF + θθF ∗ + iθσ θ∂ (φ φ∗)+ µ − 1 µ 1 µ 1 + θθθσ ∂µψ + θθσ θ∂µψ + θθθθ2(φ + φ∗) . √2 √2 4 For the components one gets

C C + φ + φ∗ −→ χ χ i√2ψ −→ − S S i√2F −→ − A A i∂ (φ φ∗) µ −→ µ − µ − λ λ −→ D D . −→ Notice that the transformations for the lowest components C, χ and S are purely algebraic (no derivatives involved), so they can always be used to put these ﬁelds to zero. This particular choice of gauge is known as Wess–Zumino (WZ) gauge. Subtleties related to the choice of the Wess–Zumino gauge will be discussed in chapter 3. It is important to note that in this gauge one has 1 V (x, θ, θ) = θσµθA (x)+ iθθθλ(x) iθθθλ(x)+ θθθθD(x) − µ − 2 1 V 2(x, θ, θ) = θθθθA (x)Aµ(x) −2 µ V n(x, θ, θ) = 0 , n 3 . ∀ ≥ Fixing the Wess–Zumino gauge still leaves an arbitrary Abelian gauge freedom on

Aµ. From the vector superﬁeld it is straightforward to construct a superﬁeld containing

the field strength for the field Aµ: it is defined as 1 W (x, θ, θ)= DDD V (x, θ, θ) (1.25) α −4 α

and is a spinor superﬁeld. Besides Wα it is useful to introduce 1 W (x, θ, θ)= DDD V (x, θ, θ) . (1.26) α −4 α˙ They are respectively chiral and antichiral:

Dα˙ Wα =0 ,DαW α˙ =0 1.4 Supersymmetric models 17

and gauge invariant

W [V +Φ+Φ†]= W [V ] .

Gauge invariance of Wα allows to compute its component expansion in the Wess– Zumino gauge; in terms of yµ one gets

i W (y, θ) = iλ(y)+ δ βD (σµσν ) β(∂ A (y) ∂ A (y)) θ + α − α − 2 α µ ν − ν µ β µ α˙ +θθσαα˙ ∂µλ (y)= (1.27) i α˙ = iλ(y)+ δ βD (σµσν) βF (y) θ + θθσµ ∂ λ (y) − α − 2 α µν β αα˙ µ and analogously for W α.

1.4 =1 supersymmetric models N The superfield formalism allows a straightforward construction of supersymmetric actions. The supersymmetry transformation of the (θθ) θθ component (F -component) of a (anti) chiral superfield is a total derivative, as can be deduced by specializing (1.21) to the case of a chiral superfield, so that an action of the form 3

4 2 2 S = d x d θ Φ+ d θ Φ† Z Z Z is automatically supersymmetric for an arbitrary chiral Φ. The same situation holds for the θθθθ component (D-component) of a vector superﬁeld yielding the supersymmetric expression

S = d4x d4θ V , Z Z for all V ’s such that V † = V . Requirement of invariance under the group of automorphisms allows to further restrict the possible couplings in the action for supersymmetric models; in this context the symmetry under transformations of the group of automorphisms is referred to as R-symmetry.

3The properties of Grassmannian integration and the conventions here employed are summarized in appendix A. 18 Chapter 1. Rigid 4d supersymmetry

1.4.1 Supersymmetric chiral theories: Wess–Zumino model

The action for the Wess–Zumino model [18] is constructed in terms of only chiral superﬁelds and reads

4 4 2 1 S = d x d θ Φ†Φ + d θ m Φ Φ + i i 2 ij i j Z Z 1 + g Φ Φ Φ + λ Φ + h.c. . (1.28) 3 ijk i j k i i In terms of component ﬁelds it becomes

4 µ µ 1 S = d x ∂ φ∗∂ φ iψ σ ∂ ψ + F ∗F + m (φ F ψ ψ )+ − µ i i − i µ i i i ij i j − 2 i j Z +g (φ φ F ψ ψ φ )+ λ F + h.c. . (1.29) ijk i j k − i j k i i The action of the Wess–Zumino model is invariant under the supersymmetry transformations

δηφi = √2ηψi µ δηψi = i√2σ η∂µφi + √2ηFi µ δηFi = i√2ησ ∂µψi

The action (1.29) does not contain derivatives of the ﬁelds Fi. They are auxiliary ﬁelds: their equations of motion are algebraic and can be used to eliminate them

from the action. Solving the equations for Fi ∂ L = Fk + λk∗ + mik∗ φi∗ + gijk∗ φi∗φj∗ =0 ∂Fk∗ ∂ L = Fk∗ + λk + mikφi + gijkφiφj =0 , ∂Fk yields the Lagrangian

µ µ 1 1 = iψ σ ∂ ψ ∂ φ∗∂ φ m ψ ψ m∗ ψ ψ + L − i µ i − µ − 2 ij i j − 2 ij i j g ψ ψ φ g∗ ψ ψ φ∗ (φ ,φ∗) , − ijk i j k − ijk i j k − V i i where the potential is given (for λ = 0) by the expression V i

=(F ∗F )=(m φ + g φ φ )(m∗ φ∗ + g∗ φ∗ φ∗ ) . (1.30) V k k ik i ijk i j mk m mnk m n 1.4 Supersymmetric models 19

This scalar potential in particular contains a φ4 interaction and the mass terms for the scalars φk; actually the Wess–Zumino model is the most general renormalizable supersymmetric theory involving only chiral superﬁelds. The action (1.29) is the most general renormalizable supersymmetric action that can be constructed in terms of only chiral superﬁelds. More generally one can consider an invariant action of the form

4 4 2 S = d x d θ Φi†Φi + d θ W(Φ) + h.c. , Z Z Z where W(Φ) is a holomorphic function of Φ called superpotential. In this case the scalar potential becomes

∂W(Φ) 2 (Φ) = . V ∂Φ

The Wess–Zumino model can be constructed in terms of unconstrained superﬁelds 3 using the property D3 = D = 0 and deﬁning

2 2 Φ= D Ψ Φ† = D Ψ† ,

The unconstrained action that reduces to (1.28), for the case of one single ﬁeld Φ, is

4 4 2 2 2 2 S = d xd θ (D Ψ†)(D Ψ) 2m(ΨD Ψ+Ψ†D Ψ+ − Z n 4g 2 2 2 2 Ψ(D Ψ) +Ψ†(D Ψ†) . (1.31) − 3! h i The equivalence follows using the identity

1 2 d4x d2θ Ψ= d4x d4θ D Ψ . −4 Z Z Z Z 1.4.2 Supersymmetric gauge theories

Gauge invariant interactions can be constructed in terms of vector superfields and more precisely in terms of the superfield Wα defined in section 1.3, which contains the gauge field strength among its components. Given a chiral superfield Φ and a vector one V an Abelian gauge theory is obtained as follows. The gauge transformation for the chiral superfield is

iqkΛ Φk′ = e− Φk , with Dα˙ Λ=0 † iqkΛ Φk† ′ = e Φk† , with DαΛ† =0 , 20 Chapter 1. Rigid 4d supersymmetry

where qk denotes the charge of the superfield Φk. The gauge parameter must be a chiral superfield Λ. Supplementing these transformations with the following one for the vector superfield V

V ′ = V + i(Λ Λ†) , − an invariant action can be written in the form

4 2 1 α 2 1 α˙ 4 q V S = d x d θ W W + d θ W W + d θ Φ†e i Φ + 4 α 4 α˙ i i Z Z Z Z 1 1 + d2θ m Φ Φ + g Φ Φ Φ + h.c. (1.32) 2 ij i j 3 ijk i j k Z Invariance under the above defined transformations is immediately verified. Notice that the action (1.32) is non-polynomial: this a general feature of supersymmetric gauge theories and is a consequence of the adimensionality of the vector superfield V . Choosing the Wess–Zumino gauge reduces (1.32) to a polynomial form since it implies V 3 = 0. This choice allows one to prove the renormalizability of the model. The vector superfield V contains the gauge dependent components C, χ and S that can be put to zero by choosing the Wess–Zumino gauge as well as the real scalar D. The latter is an auxiliary field since the corresponding equations of motion obtained from (1.32) are purely algebraic. This construction can be generalized to the non Abelian case [19, 20]. For a

generic non Abelian compact gauge group G the ﬁelds Φ, Φ† and V are matrices.

a a a † † (Φ)ij = TijΦa , Φ ij = TijΦa , Vij = TijVa , with generators T a of G satisfying

a b ab a b ab c tr(T T )= C(r)δ [T , T ]= if cT ,

C(r)= dr being the Dynkin index of the representation r. The gauge transformation takes the form

iΛ iΛ† Φ′ = e− Φ , Φ† ′ = Φ†e

and for the vector superﬁeld

V ′ iΛ† V iΛ V V ′ : e = e− e e . (1.33) −→ 1.4 Supersymmetric models 21

It can be proved that in this case as well it is possible to choose the Wess–Zumino gauge in which V 3 = 0. For inﬁnitesimal gauge transformations use of the Becker–Hausdorﬀ’s formula allows to write

δV = V ′ V = i (Λ+Λ† + coth( )(Λ Λ†) , (1.34) − LV/2 LV/2 − where (B) = [A, B] is the Lie derivative and (1.34) is a compact form to be LA understood in terms of the power expansion of coth( ). L The non Abelian generalization of the ﬁeld strength superﬁeld Wα is

1 V V W = DDe− D e . α −4 α

In the Wess–Zumino gauge the non Abelian Wα has the same form as in (1.27) with F = ∂ A ∂ A + ig[A , A ] . µν µ ν − ν µ µ ν

The transformation law of Wα induced by (1.33) is easily obtained and reads

iΛ iΛ W W ′ = e− W e , α −→ α α so that while in the Abelian case Wα was gauge invariant for a non Abelian gauge group G it turns out to be covariant. The most general renormalizable action for interacting chiral and vector super- ﬁelds is therefore

4 1 1 2 α 2 α˙ 4 V V S = d x tr d θ W Wα + d θ W α˙ W + d θ e− Φ†e Φ+ dr 16g2 Z Z Z Z 1 1 + d2θ m Φ Φ + g Φ Φ Φ + h.c. . (1.35) 2 ij i j 3 ijk i j k Z The corresponding component ﬁeld expansion in the Wess–Zumino gauge for the case mij =0, gijk = 0 is

4 1 a aµν a µ a 1 a a µ S = d x F F iλ σ λ + D D φ† φ+ −4 µν − Dµ 2 −Dµ D Z µ a a a a a a iψσ ψ + F †F + i√2g φ†T ψλ λ T φψ + gD φ†T φ , (1.36) − Dµ − where the correct factors of the coupling constant g have been recovered substituting in (1.35) V 2gV . In (1.36) denotes the ordinary covariant derivative −→ Dµ φ = ∂ φ + igAa T aφ Dµ µ µ ψ = ∂ ψ + igAa T aψ Dµ µ µ λa = ∂ λa + igf a Ab λc . Dµ µ bc µ 22 Chapter 1. Rigid 4d supersymmetry

Eliminating the auxiliary ﬁelds F and D through the equations of motion produces the scalar potential given by (1.30) plus the so called “D-term”

1 2 2 = g [φ†,φ] . VD 8 The action (1.36) can be proved to be invariant under the transformations

δξφ = √2ξψ δ ψ = i√2σµξ φ + √2ξF ξ Dµ µ a a δξF = i√2σ ξ µψ +2igT φξλ a D δ Aa = iλ σ ξ + iξσ λa (1.37) ξ µ − µ µ a µν a a δξλ = σ ξFµν + iξD a δ Da = ξσµ λ λaσµξ . ξ − Dµ −Dµ The Wess–Zumino condition is not supersymmetric so that it not only (partially) breaks gauge invariance, but it also breaks supersymmetry, that is recovered through a non linear realization given by equations (1.37). These transformations are more precisely a mixture of supersymmetry and the residual non Abelian gauge symmetry. The choice of the Wess–Zumino gauge leads to subtleties at the quantum level, as will be discussed in chapter 3. The super-gauge invariant action (1.35) can be generalized by introducing a com- plexiﬁed coupling constant θ 4πi τ = + 2π g2 obtaining, in the pure super Yang–Mills case, the analytic expression 1 S = Im τ d4xd2θ trW αW = 8π α Z 1 1 1 = d4x tr F F µν iλσµ λ + D2 + g2 −4 µν − Dµ 2 Z θ d4x trF F˜µν , −32π2 µν Z ˜µν 1 µνρσ where F = 2 ε Fρσ.

1.5 Theories of extended supersymmetry

The most straightforward way to construct multiplets for theories with extended ( > 1) supersymmetry is to assemble together =1 multiplets in such a way as to N N 1.5 Extended supersymmetry 23

obtain a field content corresponding to the representations of section 1.2. It turns out that in general this program can be achieved only on-shell; in other words it is not possible to realize -extended supersymmetry in a closed form through an off-shell N multiplet of fields. With the exception of =2 supersymmetric Yang–Mills theories N closure of the algebra without imposing the equations of motion would require an infinite set of auxiliary fields. This problem is related to the presence of central charges in the supersymmetry algebra for > 1. N As in the =1 case the notion of superspace will be introduced to start with and N then explicit realizations in specific models will be considered.

1.5.1 Extended superspace: a brief survey

The aim of this section is to give a brief account of ideas beyond the superspace formulation for theories with extended supersymmetry, no claim of completeness is made. In the following extended superspace will not be used in explicit calculations, but it will rather be invoked to justify some statements. A detailed and self-contained review of extended superspace is given in [21]. The general construction of superspace as a coset space introduced in section 1.3 can be generalized to deﬁne extended superspace as the coset K=SP∗/L, where SP∗ is the extended super Poincar´egroup generated by the general algebra (1.1), (1.3) and L is the Lorentz group SO(3,1). The generic element of SP∗ can be written

α˙ i(aµP +ηαQi +ηi Q +brZ + 1 w M µν ) g = e µ i α α˙ i r 2 µν ,

r where Zij = ΩijZr are the central charges of the supersymmetry algebra and in this case there are anticommuting generators Qi . Elements of the coset are obtained N α setting wµν = 0 and can be expressed in the form

µ α i i α˙ r k = ei(x Pµ+θi Qα+θα˙ Qi +ζ Zr) . (1.38)

Consequently they are parametrized by the enlarged set of coordinates

i j zm =(xµ, θα, θα˙ ,ζr) . (1.39)

Repeating the steps of section 1.3 allows to obtain the following supersymmetric 24 Chapter 1. Rigid 4d supersymmetry

coordinate transformations

µ µ α µ αi˙ α µ αi˙ x′ = x + iη σ θ iθ σ η i αα˙ − i αα˙ α α α θi′ = θi + ηi αj˙ αj˙ θ′ = θ + ηαj˙ (1.40) r r α r ij i αj˙ r ζ′ = ζ + iηi θαj(Ω ) + iηα˙ θ (Ω )ij .

µ i Under the action of the central charges x , θα and θαj˙ do not transform while

r r r ζ′ = ζ + b . (1.41)

Superﬁelds are functions deﬁned on the manifold parametrized by zm

i j F = F (xµ, θα, θα˙ ,ζr) . (1.42)

The action of the super Poincarégroup on superfields is defined through

R(g) F (z)= F (z′)

where z′ is given in (1.40) and R(g) acts as the left regular representation. Diﬀerential operators generating the inﬁnitesimal transformations now read

ℓµ = ∂µ

i ∂ µ αi˙ r ij ∂ ℓα = α + iσαα˙ θ ∂µ + i(Ω ) θαj r ∂θi ∂ζ ∂ α µ j r ∂ ℓαi˙ = αi˙ + iθi σαα˙ ∂µ + iθα(Ω )ij ∂θ ∂ζr ∂ ℓr = ∂ζr ˙ 1 β µν α ∂ 1 β µν α˙ ∂ ℓµν = (xµ∂ν xν ∂µ) θ σβ + θ σ ˙ α˙ − − − 2 ∂θα 2 β ∂θ ∂ αi˙ ∂ ℓ = θαBi θ Bi ∗ . a − i aj ∂θα − aj α˙ j ∂θj Relation to component ﬁeld multiplets is established by Taylor expanding in the θ, θ as well as in the ζ coordinates. Notice that the ζ’s are “bosonic” coordinates, so that the expansion is a truly inﬁnite series

(0) (1) (r) F (x, θ, θ,ζ)= F (x, θ, θ)+ F(r) (x, θ, θ) ζ + ... , (1.43) 1.5 Extended supersymmetry 25

where

F (0)(x, θ, θ) = F (x, θ, θ, 0)

(1) ∂ F(r) (x, θ, θ) = F (x, θ, θ,ζ) . ∂ζr ζ=0

“Super-covariant” derivatives generalizing those deﬁned in section 1.3 are given by

Dµ = ∂µ

i ∂ µ αi˙ r ij ∂ Dα = α + iσαα˙ θ ∂µ i(Ω ) θαj r ∂θi − ∂ζ ∂ α µ r j ∂ Dαi˙ = αi˙ iθi σαα˙ ∂µ i(Ω )ijθα˙ (1.44) −∂θ − − ∂ζr ∂ D = . r ∂ζr

Just as in the case of =1 superspace constraints defining superfields correspond- N ing to irreducible representations of supersymmetry are implemented through the covariant derivatives (1.44). In general constrained superfields still depend on the coordinates ζ, so that the component supermultiplets contain an infinite number of fields; the only exception is the =2 vector superfield defined by the condition N ∂ D V (x, θ, θ,ζ)= V (x, θ, θ,ζ)=0 . (1.45) r ∂ζr

As will be discussed in the next subsection the component fields associated to this superfield are those defining the =2 supersymmetric Yang–Mills model. N > 1 irreducible on-shell representations constructed in section 1.2 contain a N 2 finite number of states (2 N in the massive case and 2N in the massless case). The reason for the appearance here of an infinite number of component fields is that closure of the supersymmetry algebra off-shell in the presence of central charges requires infinitely many auxiliary fields, the only exception being the =2 super N Yang–Mills multiplet related to the superfield in equation (1.45). The =2 supersymmetry algebra contains in principle two central charges, but N one of them can always be eliminated through a chiral rotation, so that the only relevant anticommuting relation is

Di ,Dj =2ε εijD . { α β} αβ ζ 26 Chapter 1. Rigid 4d supersymmetry

One can consider a superﬁeld W satisfying

i Dα˙ W =0 ,DζW =0 . (1.46)

A multiplet of ﬁelds can be obtained taking the θ = 0 component of the ﬁelds

i αi j i W,DαW,D DαW,DαDβiW , i j k i j k l DαDβDγ W,DαDβDγ DδW . (1.47)

Exploiting the conditions (1.46) in (1.47) does not produce an irreducible multiplet. A further reduction is necessary which is achieved by imposing the reality condition

αi j i αj˙ D DαW = Dα˙ D W .

This gives rise to a multiplet containing the ﬁelds

i ij D , χα , C , Fµν , (1.48)

where D and Cij = Cji are real scalars, χi Weyl spinors and F = ∂ A ∂ A is α µν µ ν − ν µ the field strength for a real vector field A . These are the fields of the =2 super µ N Yang–Mills model: they sum up a chiral and a vector =1 multiplet. N An example of a =2 multiplet possessing central charge arises if one considers N a superfield satisfying 1 Di Φ = δi Dk Φ α j 2 j α k Dαi˙ Φj + Dαj˙ Φi =0 . (1.49)

Using these constraints a multiplet of ﬁelds can be obtained taking the θ = 0 component of the ﬁelds

k k Φ ,DαΦk , Dα˙ Φk ,DζΦ .

Some D-algebra manipulations allow to construct a supermultiplet containing two Weyl spinors and four complex scalars

ψα , χα , φi , Fi . (1.50)

This multiplet is known as hypermultiplet and can be viewed as the combination of two =1 chiral multiplets. It will be shown to describe =2 matter in supersymmetric N N gauge theories. 1.5 Extended supersymmetry 27

In general the constraints that need to be imposed on > 1 superfields cannot N be explicitly solved for the whole superfield; in other words considering all the θ- components yields a set of non independent fields. The lack of an explicit solution of the constraints does not allow to exploit the potentiality of the superspace formalism for extended supersymmetry, so that it does not prove so powerful a tool as in the =1 case. N These problems were partially solved in [22] by the introduction of harmonic superspace. It is based on a parametrization of =2 superspace as a homogeneous space N that leads to different bosonic coordinates associated to the central charges. Use of such coordinates allows to construct =2 hypermultiplets in terms of unconstrained N superfields. A generalization of this approach leading to the so called analytic superspace has been proposed in order to give a manifestly supersymmetric description of =4 theories (see the review [21] for the details), however it only works on-shell. N In the following all the explicit calculations using superfield formalism will be carried out within the framework of =1 superspace. N 1.5.2 > 1 supersymmetric models N In this section only examples of =2 theories will be discussed, the general properties N of =4 models, that are the main subject of this work, will be studied in detail in N next chapter.

=2 supersymmetric Yang–Mills model. N The superfield formulation of =2 pure supersymmetric Yang–Mills models is a N straightforward generalization of the = 1 case because it is constructed in terms N of a =2 superfield satisfying the constraint (1.45), which implies no dependence on N the central charges. Since no central charges are present a full off-shell formulation is possible. Superspace in this case is parametrized by

i ˜ ˜ zm =(xµ, θα, θαj˙ )=(xµ, θα, θα˙ , θα, θα˙ ) The starting point is a chiral superﬁeld Ψ deﬁned by

Dα˙ Ψ=0 , D˜ α˙ Ψ=0 (1.51)

There is a natural choice of variables to describe Ψ, namely ˜ ˜ y˜µ = xµ + iθσµθ + iθσµθ 28 Chapter 1. Rigid 4d supersymmetry

Ψ can be expressed in terms ofy ˜ as

α Ψ(˜y, θ) =Φ(˜y, θ)+ i√2θ˜ Wα(˜y, θ)+ θ˜θG˜ (˜y, θ) ,

where Φ and G are =1 chiral superﬁelds and W is a chiral spinor superﬁeld. N α To construct a non Abelian gauge theory Ψ is taken to be a matrix

a Ψij = TijΨa .

A =2 supersymmetric action can then be written as N 1 1 S = Im τ d4x d2θd2θ˜tr Ψ2 , (1.52) 4π 2 Z Z where

2 α Ψ = W W 2 + 2GΦ 2 . θ2θ˜2 α|θ |θ =2 super Yang–Mills is obtained by further constraining the chiral superﬁeld G. N This is achieved by requiring Ψ to satisfy

αi j i αj˙ D DαΨ= Dα˙ D Ψ† .

Solving the constraint provides for G the following expression

2 V (˜y iθσθ,θ) G = d θ Φ†(˜y iθσθ, θ)e − . − Z Substituting into the general form (1.52) gives the action of =2 super Yang–Mills N theory. It can be viewed as a particular =1 supersymmetric gauge theory coupled N to chiral matter, namely it corresponds to the case of one single flavor in the adjoint representation of the gauge group. Supersymmetry does not allow a superpotential for the model. Integration over the θ variables gives the component expansion. The superfield Ψ is a singlet under the SU(2) R-symmetry group. The resulting component multiplet obtained eliminating the auxiliary fields in the Wess–Zumino gauge contains

a vector Aµ and a complex scalar φ that are SU(2) singlets and two Weyl spinors (ψ,λ) that transform in the 2 of SU(2). As already noticed, this field content is what would be obtained putting together a vector and a chiral =1 superfield both in the N adjoint of the gauge group. The most general =2 action that can be constructed in terms of the superfield N Ψ of equation (1.51) is 1 S = Im d4x d2θd2θ˜ (Ψ) , 4π F Z Z 1.6 Quantization of SUSY theories 29

where is a holomorphic function of Ψ called prepotential. F

=2 matter. N The =2 super Yang–Mills model can be generalized introducing the coupling to N matter described by hypermultiplets. Hypermultiplets are associated to superﬁelds satisfying (1.49). A kinetic term for these superﬁelds can be written in the form

4 αi j i βk k j SK = d x D Dα Φ †D DβΦ . Z Explicit dependence on the central charge does not allow a manifestly =2 formu- N lation without resorting to harmonic superspace. Anyway an on-shell action can be constructed in terms of = 1 superﬁelds. Denoting the two chiral superﬁelds in the N hypermultiplet by Q and Q˜ the action in =1 language is given by N

4 4 2V 2V 2V S = d x d θ Q†e Q + Q˜†e Q˜ + Φ†e Φ + Z Z 1 h i + Im τ d2θ W αW + d2θ √2Q˜ΦQ + m Q˜ Q + h.c. . (1.53) 4π α i i i Z Z The action (1.53) contains the ordinary kinetic terms, a mass term for chiral fields in the hypermultiplet and a superpotential that couples Q, Q˜ and Φ. Notice that no potential term coming from a self interaction of the fields Q and Q˜ is allowed by =2 supersymmetry. This follows from the observation that Q and Q˜ are assembled N into a superfield Φi transforming in the 2 of SU(2) R-symmetry and there exists no trilinear invariant in SU(2).

1.6 Quantization of supersymmetric theories

The functional approach to the quantization of field theories, based on the construction of a generating functional for the time-ordered products of fields, can be applied with no further difficulty to supersymmetric theories in their component field formulation. However, although this formulation allows to extract all the peculiar properties of supersymmetric theories at the quantum level, it makes rather obscure how such properties are a direct consequence of supersymmetry. On the contrary a quantization procedure can be carried out directly in superspace in a step by step manifestly supersymmetric fashion. This approach, that allows to construct directly 30 Chapter 1. Rigid 4d supersymmetry

Green functions for the superfields, has proved extremely powerful and it is perhaps the most useful application of superspace techniques. Super Feynman rules for the computation of Green functions of the superfields were first proposed in [14] and then reformulated in a more powerful way in [23]. This latter formulation will be reviewed here. Once the Green functions for the superfields are known contact with the “physical” component field formulation is established by further integrating the fermionic coordinates (θ, θ) to extract correlators for the various components. The discussion will focus on the quantization of =1 superfields; it is also possible N to generalize the method to -extended superspace, e.g. to harmonic superspace, N but the =1 formulation appears to be the most suitable for application to explicit N calculations, even in the extended case.

1.6.1 General formalism

In ordinary quantum ﬁeld theories the path integral quantization is based on the

construction of a generating functional for the Green functions. Denoting by φi the generic ﬁelds of the model and by S[φ] the classical action the generating functional is deﬁned as

Z[J]= N [ φ] eiS[φ]+i dxφi(x)Ji(x) , D Z R

where N is a normalization such that Z[0] = 1 and Ji(x) are external sources asso-

ciated to the ﬁelds φi(x). Green functions are obtained taking functional derivatives

with respect to the sources Ji. This approach has a natural extension that leads to a generating functional for the Green functions of the superﬁelds in a supersymmetric theory. The general form of the action of a supersymmetric theory involves, in =1 N language, chiral and vector superﬁelds and can be written

4 4 4 2 S = d xd θK(Φ, Φ†,V )+ d xd θ W[Φ] + h.c. . (1.54) Z Z To construct a generating functional classical sources for the different superfields must be introduced, that are in turn superfields. Since functional derivatives are to be taken with respect to the classical sources, such sources must be arbitrary. A problem arises with chiral superfields as sources coupled to them must themselves 1.6 Quantization of SUSY theories 31

be chiral. This difficulty can be solved by using a suitable definition of functional derivatives with respect to chiral superfields. For a generic superfield F (x, θ, θ) functional derivatives are defined according to the rule

δF (x′, θ′, θ′) = δ4(x x′)δ4(θ θ′) , δF (x, θ, θ) − − where the δ function of fermionic variables is to be interpreted as

2 2 δ (θ θ′)= δ (θ θ′)δ (θ θ′)=(θ θ′) (θ θ′) , 4 − 2 − 2 − − − so that

4 d θ δ (θ θ′)F (x, θ, θ)= F (x, θ′, θ′) 4 − Z and

δ 4 4 d x′d θ′ F (x′, θ′, θ′)G(x′, θ′, θ′)= G(x, θ, θ) . (1.55) δF (x, θ, θ) Z For chiral superﬁelds Φ one deﬁnes in turn

δΦ(x′, θ′, θ′) 1 2 = D δ4(x x′)δ4(θ θ′) . δΦ(x, θ, θ) −4 − −

In this way the same result as in equation (1.55) is reproduced

δ 4 2 d x′d θ′ Φ(x′, θ′, θ′)Φ(˜ x′, θ′, θ′)= δΦ(x, θ, θ) Z 4 2 1 2 = d x′d θ′ D δ (x x′)δ (θ θ′)Φ(˜ x′, θ′, θ′)= −4 4 − 4 − Z 4 4 = d x′d θ′δ (x x′)δ (θ θ′)Φ(˜ x′, θ′, θ′)= Φ(˜ x, θ, θ) , 4 − 4 − Z 2 where in the last line it was used the property that d2θ and 1 D are equivalent under − 4 space-time integration, as can be proved by direct calculation by an integration by parts. Given these deﬁnitions of functional derivatives the generating functional can be written as

† † † iS[Φ,Φ ,V ]+i(Φ,J)+i(Φ ,J )+i(V,JV ) Z[J, J †,J ]= N [ Φ Φ† V ] e , (1.56) V D D D Z 32 Chapter 1. Rigid 4d supersymmetry

where S is the action (1.54), J (J †) is a chiral (antichiral) source, JV is real and

(Φ,J) = d4xd2θ Φ(x, θ, θ)J(x, θ, θ) , Z 4 2 (Φ†,J †) = d xd θ Φ†(x, θ, θ)J †(x, θ, θ) , Z 4 4 (V,JV ) = d xd θV (x, θ, θ)JV (x, θ, θ) . Z Green functions can then be calculated by the formula

(z ,...,z , z ,... ,z , z ,... ,z )= Gn 1 i i+1 j j+1 n = 0 T Φ(z ) . . . Φ(z )Φ†(z ) . . . Φ†(z )V (z ) ...V (z ) 0 = (1.57) h | { 1 i i+1 j j+1 n }| i δnZ[J, J ,J ] = ( i)n † V . − δJ(z1) ...δJ(zi)δJ †(zi+1) ...δJ †(zj)δJV (zj+1) ...δJV (zn) † J=J =JV =0

A generating functional for the connected Green functions can be obtained by taking the logarithm of Z

W [J, J †,J ]=( i) log Z[J, J †,J ] . V − V The quantum effective action Γ that acts as a generating funct ional for the one particle irreducible Green functions is obtained through the Legendre transform. Defining the “classical” fields δW δW δW Φ=˜ , Φ˜ † = , V˜ = , (1.58) δJ δJ † δJV Γ is given by ˜ ˜ ˜ Γ[Φ, Φ†, V ]= W [J, J †,JV ] (J, Φ)+(J †, Φ†)+(JV ,V ) ˜ ˜ † ˜ , − (Φ,Φ ,JV ) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ where on the right hand side J = J[Φ, Φ†, V ], J † = J †[Φ, Φ†, V ] and J V = JV [Φ, Φ†, V ] come from the inverting (1.58). The generating functional Z can be expressed in terms of the corresponding func-

tional in the free theory, Z0. The general action (1.54) can be divided up into a free part and an interacting part

S[Φ, Φ†,V ]= S0[Φ, Φ†,V ]+ Sint [Φ, Φ†,V ] ,

where S0 is quadratic in the ﬁelds and contains the kinetic and mass terms. Then one can write

δ δ δ iSint δJ , † , δJ Z[J, J †,JV ]= e δJ V Z0[J, J †,JV ] , (1.59) h i 1.6 Quantization of SUSY theories 33

where

† † † iS0[Φ,Φ ,V ]+i(Φ,J)+i(Φ ,J )+i(V,JV ) Z [J, J †J ]= N [ Φ Φ† V ] e 0 V D D D Z δ δ δ iSint [ δJ , † , δJ ] and e δJ V is to be understood as a power expansion. Since S0 is quadratic

Z0 is a Gaussian integral that can be explicitly computed. Writing

i 4 4 Φ Z0[J, J †,JV ] = exp d xd θ (Φ, Φ†) Φ + −2 M Φ† Z i 4 4 + d xd θV V + i(Φ,J)+ i(Φ†,J †)+ i(V,J ) (1.60) 2 MV V Z one gets

i 8 8 J(z′) Z0[J, J †,JV ] = exp d zd z′ (J(z),J †(z))∆Φ(z, z′) + 2 J †(z′) Z i 8 8 + d zd z′ J (z)∆ (z, z′)J (z′) , (1.61) 2 V V V Z where superﬁeld propagators, ∆Φ and ∆V , have been introduced.

1.6.2 Superﬁeld propagators

To compute the propagators the free action must be put in a canonical quadratic form. Since the computation leads to different kinds of difficulties for chiral and vector fields the two cases will be considered separately.

Chiral superﬁeld propagator.

The general form of the free action is

4 4 2 1 2 1 S = d x d θ Φ†Φ + d θ m Φ Φ + d θ m Φ†Φ† = 0 i i 2 ij i j 2 ij i j Z Z Z Z 4 4 1 DD DD = d xd θ Φ†Φ m Φ Φ + Φ† Φ† , (1.62) i i − 8 ij i 2 j i 2 j Z 2 where the last line follows from the equivalence d2θ 1 D2, d2θ 1 D , valid ↔ − 4 ↔ − 4 under a space-time integration, and the action of the projection operators

2 1 D2D P = , P Φ† = Φ† , P Φ=0 1 16 2 1 1 2 1 D D2 P = , P Φ = Φ , P Φ† =0 . (1.63) 2 16 2 2 2 34 Chapter 1. Rigid 4d supersymmetry

Hence the action can be rewritten as

1 4 4 Φj S0 = d xd θ Φi, Φi† ij , (1.64) 2 M Φj† Z where 1 mij 4 2 DD δij ij = − . 1 mij M δij DD ! − 4 2 Considering for simplicity the case of one single ﬂavor and taking into account the

source terms, the exponent of the Gaussian integral that yields Z0 turns out to be 1 D2 4 4 1 Φ 4 2 0 J i d xd θ Φ, Φ† + Φ, Φ† − 2 . 2 M Φ† 0 1 D J † Z " 4 2 ! # − A simple but lengthy calculation using D-algebra and properties of the projection

operators allows to write Z0 in the form of equation (1.61), where the chiral superﬁeld propagator, known as Grisaru–Roˇcek–Siegel propagator [23], reads

m D2 1 4 2 1 ∆Φ(x, θ, θ; x′, θ′, θ′) = 2 δ8(z z′) (1.65) 2 m2 1 m D − − 4 2 ! and δ (z z′)= δ (x x′)δ (θ θ′). 8 − 4 − 4 −

Vector superﬁeld propagator.

The free ﬁeld action for a vector superﬁeld is

1 1 α˙ S = d4x d2θ W αW + d2θ W W + d4θ m2V 2 . (1.66) 0 4 α 4 α˙ Z Z Z Z No massive vector field will enter the models considered in the following so m will be put to zero from the beginning. The kinetic operator in (1.66) has zero-modes so that just like in ordinary gauge theories a gauge fixing procedure àla Faddeev–Popov is necessary. This requires the introduction of a gauge fixing term in (1.66) and a further integration over the Faddeev–Popov ghost fields will appear. The gauge-fixed action thus becomes

S = S0 + SGF + SFP = 1 1 α˙ = d4xd4θ W αW δ (θ)+ W W δ (θ) + 4 α 2 4 α˙ 2 Z ξ 2 + d4xd4θ (D V )(D2V ) + −8 Z 4 4 +i d xd θ (C′ + C′) V (C + C) + coth V (C C) , (1.67) L 2 L 2 − Z h i 1.6 Quantization of SUSY theories 35

where C, C′ are chiral superfields for the ghosts and no source term has been introduced for such fields. The gauge fixing term here introduced corresponds to a class of gauges that for the component field Aµ interpolate between the Lorentz gauge (ξ = ) and the Fermi–Feynman gauge (ξ = 1). ∞ The free propagators for the ghosts are exactly the same as for ordinary chiral superfields. Furthermore there is a non polynomial interaction with the vector superfield coming from the expansion of coth( ). LV/2 The vector superfield propagator is calculated from the quadratic part of the action

S = S0 + SGF = = d4xd4θ V [ 2P ξ(P + P )2] V = { − T − 1 2 } Z = d4xd4θV V . (1.68) MV Z Introducing the source term for V the generating functional becomes

i V V V +i(V,JV ) Z[J ]= [ V ] e M . V D Z R The Gaussian integral can now be performed yielding a result of the form (1.61), where the vector superﬁeld propagator is 1 α ∆ (x, θ, θ; x′, θ′, θ′) = P (P + P ) δ (z z′)= V −2 T − 2 1 2 8 − 1 = [1+(α 1)(P + P )] δ (z z′) , (1.69) −2 − 1 2 8 − 1 with α = ξ .

1.6.3 Feynman rules

Once the free ﬁeld propagators are known the perturbative expansion for the Green functions of the superﬁelds can be derived from the generating functional in the form (1.59) that yields

(z ,...,z , z ,...,z , z ,... ,z )= Gn 1 i i+1 j j+1 n = 0 T Φ(z ) . . . Φ(z )Φ†(z ) . . . Φ†(z )V (z ) ...V (z ) 0 = h | { 1 i i+1 j j+1 n }| i δ δ δ δ δ δ = ...... δJ(z1) δJ(zi) δJ †(zi+1) δJ †(zj) δJV (zj+1) δJV (zn) · k ∞ ( i)k δ δ δ − Sint , , Z0[J, J †,JV ] . (1.70) · k! δJ δJ † δJV k=0 † J=J =JV =0 X

36 Chapter 1. Rigid 4d supersymmetry

A super Feynman graph technique can be developed which is a straightforward generalization of what is done in ordinary ﬁeld theories. Super Feynman rules for the evaluation of one particle irreducible Green functions can be derived given the above generating functional. The rules can be summarized as follows.

To each external line is associated the corresponding superﬁeld. • At each vertex there is an integration over the whole superspace d4xd4θ; factors, • as for example powers of the coupling constant, to be associated to the vertices, are read directly from the action.

2 A factor of 1 D or 1 D2 acting on the internal chiral or antichiral lines • − 4 − 4 respectively must be included at each vertex. Such factors come from the deﬁnition of the functional derivatives. At purely chiral (antichiral) vertices one 2 factor of 1 D ( 1 D2) must be suppressed, because it is used to reconstruct − 4 − 4 the measure over the whole superspace.

For internal lines the propagators previously derived are used. • The speciﬁc form of the interaction at each vertex as well as the combinatoric • analysis are worked out exactly like in ordinary perturbation theory.

The expressions for the one particle irreducible Green functions in momentum • space are obtained by Fourier transforming in the x variables, but not in the fermionic θ coordinates.

1.7 General properties of supersymmetric theories

Supersymmetry has far reaching consequences in the quantum theory both at the perturbative and at the non-perturbative level. Some general results will be brieﬂy recorded in this section. More can be found in the books and reviews quoted in the references. Some dramatic eﬀects on the divergences encountered in perturbation theory can be easily derived from the Feynman rules. Explicit calculation shows that in Feynman diagrams four covariant derivatives are associated at each vertex, with the exception of vertices involving external (anti) chiral lines that have two derivatives less. More- 2 D2 D over ΦΦ (Φ†Φ†) propagators with momentum p have an additional factor of p2 ( p2 ). 1.7 General properties 37

Simple dimensional analysis allows to compute the superﬁcial degree of divergence ds of a diagram which is

d =4L 2P +2V C E 2L , s − − − − where L is the number of loops, P the number of internal propagators, V the number of vertices, C the number of ΦΦ or Φ†Φ† propagators and E the number of external chiral or antichiral lines. Use of the topological constraint L P + V = 1 yields [24] − d =2 C E . s − − For graphs with only external V lines gauge invariance requires the presence of four D or D factors on the external lines so that the result is

d = C s − 2 For diagrams with only lines of a given chirality there are further factors of D2 or D associated to the external lines so that one obtains

d =1 C n , s − − where n is the number of external chiral or antichiral lines. This calculation allows to determine all the potentially divergent diagrams. The only possible divergences are logarithmic and can arise in diagrams with only external V lines with C =0(e.g. corrections to the vector superﬁeld propagator) and in

Φ†Φ propagators with C = 0. This shows that in supersymmetric models only wave function renormalizations may be required; there is no direct mass or coupling 2 constant renormalization, i.e. with W = mφ one has ZW = 1, but Zφ = 1 implies 1/2 6 Z = Z− = 1. m φ 6 This and other results can be obtained from the general non renormalization theorem for =1 theories [25]: Any perturbative quantum contribution to the eﬀective N action Γ can be expressed as an integral over the whole superspace and is local in the θ variables, namely

Γ = d4x ...d4x d4θG(x ,...,x ) 1 n 1 n · n X Z f [Φ, Φ†,V,D Φ, D Φ†,D V, D V,... ] . (1.71) · n α α˙ α α˙ The theorem can be easily proved using Feynman rules and successively integrating by parts the θ variables exploiting the properties of Grassmannian integration. Explicit 38 Chapter 1. Rigid 4d supersymmetry

calculations in the =4 supersymmetric Yang–Mills theory showing the mechanism N leading to this result will be presented in chapter 3, so the proof of the above statement will not be discussed here. Notice that the non renormalization theorem does not forbid in principle quantum corrections to the superpotential. It is easy to see that a contribution of the form of (1.71) can produce a correction to the superpotential [26, 27]. As an example consider a term of the form

2 D2 D D2 d4xd2θd2θ Φn = d4xd2θ Φn = d4xd2θΦn , 162 162 Z Z Z 2 D D2 where Φ is chiral. The right hand side, which follows since P1 = 162 is the projector on chiral superfields, has the form of a correction to the superpotential. This example shows that in principle an arbitrary correction to the superpotential can be generated perturbatively. This effect is related to an infrared singular behaviour that can be present only in theories containing massless fields. The example shows that the correction, which is non-local when written as an integral over the whole superspace, can be rewritten as a local term integrated over a subspace. The same construction could not be repeated if the theory does not contain massless particles, since in that 1 case one would obtain a factor of 2+m2 , which cannot be re-expressed as a subintegral. In [27] it was shown that such a correction as the one considered above actually occurs at the two-loop level in the Wess–Zumino model. In a =1 super Yang–Mills theory N coupled to the Wess–Zumino model the correction appears at one loop [28]. An alternative approach to the study of this kind of effects has been given in [29, 30]. Other non renormalization theorems hold for theories of extended supersymmetry. In particular considerations based on the so called anomalies argument allow to prove the finiteness of =2 theories beyond one loop. More precisely it can be proved that N the only perturbative contributions to the β function of extended supersymmetric theories can come from one loop diagrams. This result allows to single out a class of finite supersymmetric models and among them the =4 super Yang–Mills theory N that will be studied in detail in the following chapters. Supersymmetry puts severe restrictions on the non-perturbative properties as well. A review of the general results on the non-perturbative dynamics of supersymmetric theories is provided by [31, 32]. Many astonishing results in this context can be traced back to the properties of holomorphy of supersymmetric theories. In supersymmetric models for example the superpotential is a holomorphic function of the chiral super- 1.7 General properties 39

ﬁelds. Analogously, since the parameters of the theories (such as masses and coupling constants) can be viewed as expectation values of auxiliary ﬁelds, various quantities depend analytically on such parameters. Using holomorphy properties many exact result have been derived. In [33] an expression for the β function of supersymmetric gauge theories was proposed that is exact at the perturbative as well as at the non- perturbative level for a class of models. Also on holomorphy relies the construction of the exact solution for the low energy dynamics of a class of =2 theories proposed N by Seiberg and Witten [34]. Chapter 2

=4 supersymmetric Yang–Mills N theory

Four dimensional =4 supersymmetric Yang–Mills theory is a very special quantum N field theory. It possesses several peculiar properties that will be reviewed in this chapter. The general properties of the model are discussed here, original results in the study of =4 Yang–Mills will be presented in the following chapters. N The action of the theory was given for the first time in [35, 36] within the framework of string theory toroidal compactifications. The theory has the maximal amount of supersymmetry for a rigid supersymmetric theory in four dimensions, namely sixteen real supercharges; a larger number of supercharges would require fields of spin larger than one and therefore the inclusion of gravity. Historically the interest in this model was raised by its property of finiteness; the β function of Gell-Mann and Low has been proved to be vanishing in perturbation theory and the same is supposed to be true at the non-perturbative level. As a consequence the superconformal invariance displayed by the classical theory is believed to be preserved after quantization. Moreover =4 super Yang–Mills theory possesses exact electric-magnetic duality. N It contains beyond the elementary fields an infinite set of dyonic states which are believed to realize exactly the duality symmetry proposed by Montonen and Olive in [37]. More recently there has been a renewal of interest in =4 Yang–Mills theory N 40 2.1 Formulations of =4 SYM 41 N following the proposal made by Maldacena in [38] of a new duality relating type IIB supergravity in d + 1 dimensional anti-de Sitter space and d dimensional (super)conformal theories. This subject will be discussed in detail in chapter 5. The present chapter is organized as follows. In section 2.1 the various formulations of the model, which will be employed at different stages in the following, are reviewed. Section 2.2 presents the classical symmetries and conserved currents of the theory. The properties of finiteness are discussed in section 2.3. The final section is then devoted to electric-magnetic duality in =4 supersymmetric Yang–Mills theory. The N general aspects of electric-magnetic duality are reported separately in appendix B.

2.1 Diverse formulations of =4 supersymmetric N Yang–Mills theory

=4 supersymmetric Yang–Mills theory in four dimensions was obtained for the first N time in [35, 36] by applying the method of dimensional reduction to =1 super Yang– N Mills in ten dimensions. The latter is the low energy effective theory coming from type I superstring theory and describes a = 1 vector multiplet in ten dimensions N consisting of one real vector and one Majorana–Weyl spinor. The action is 1 1 i S = d10x tr F F ΓΛ + λΓΛD λ , (2.1) −4 ΓΛ 2 Λ Z where F = ∂ A ∂ A + ig[A , A ] and λ satisfies the Majorana and Weyl ΓΛ Γ Λ − Λ Γ Γ Λ conditions

T λ = C λ , λ = Γ λ , (10) ± 5 where C(10) is the charge conjugation operator. The action (2.1) is invariant under the supersymmetry transformations

δηAΛ = iηΓΛλ ΓΛ δηλ = ΣΓΛF η . (2.2)

In the previous equations capital Greek letters Γ and Σ denote ten dimensional matrices. Dimensional reduction ´ala Kaluza–Klein on a six dimensional torus T 6 leads to a multiplet of ﬁelds in four dimensions possessing an additional SU(4) SO(6) global ∼ 1Capital Greek indices refer to ten dimensional space-time. 42 Chapter 2. =4 super Yang–Mills theory N

symmetry, which is a direct consequence of the ten dimensional Lorentz invariance. More precisely it is a consequence of the fact that the torus T 6 has a trivial holonomy group. In general in the dimensional reduction on a Riemannian manifold of dimension k the spin connection is a SO(k) gauge field and consequently the spinors transform, upon parallel transport around a closed contractible curve, under a subgroup of SO(k), which is the holonomy group. In the case at hand of T 6 every spinor is covariantly constant, so that the whole SO(6) SU(4) becomes a global symmetry ∼ of the model. From the point of view of the =4 theory this SU(4) global symmetry N is identified with the R-symmetry group of the =4 supersymmetry algebra, as will N be discussed in the next section. In the bosonic sector the compactification gives rise to a four dimensional real vector and to six real scalars from the internal components of the ten dimensional vector. More precisely one defines

a a Aµ = Aµ µ =0, 1, 2, 3 ϕa = 1 (Aa + iAa ) B =1, 2, 3 B4 √2 B+3 B+6 a AB 1 ABC4 a a AB ∗ ϕ = 2 ε ϕC4 = ϕ A,B,C =1, 2, 3 . (2.3)

In the fermionic sector a suitable choice of the ten-dimensional Γ matrices allows to write, in Majorana notation, the 32-component Majorana–Weyl spinor λ as Lχ1 .  .  4 Lχ A T λ =   , χ˜A = C(χ ) ,  Rχ1   .   .     Rχe   4    where L and R denote the left and right chirality projection operators and C is e the four-dimensional charge conjugation operator related to the ten-dimensional one 0 1I4 by C(10) = C . In four dimensions one obtains four Majorana (or ⊗ 1I4 0 equivalently Weyl) spinors

LχA λA = A =1, 2, 3, 4 . (2.4) RχA ! At the end of the day the result of the compactiﬁcation is the = 4 multiplet e N a AB a which contains six real scalars ϕ satisfying (2.3), one real vector Aµ and four 2.1 Formulations of =4 SYM 43 N

aA Weyl spinors λα , all in the adjoint representation of the gauge group. The construction here described from the ten dimensional theory allows to derive immediately the transformation of the fields under the SU(4) global symmetry. Because of the constraint (2.3) the scalars ϕa AB are in the second rank complex self dual 6 of SU(4); aA a a the spinors λ are in the 4 while the λA transform in the 4; the gauge field Aµ is a singlet. In the following a different notation for the scalar fields will also be used, namely one can define 1 ϕi = ti ϕAB , i =1, 2,..., 6 , 2 AB AB where (ti) are Clebsch–Gordan coefficients that couple two 4’s to a 6 (these are µ six-dimensional generalizations of the four-dimensional σ αα˙ matrices). In Weyl notation the action of the four dimensional =4 theory turns out to be N 1 ←→ α˙ 1 S = d4x tr (D ϕAB)(Dµϕ ) i(λαA D/ λ ) F F µν + µ AB − 2 αα˙ A − 4 µν Z n α˙ gλαA[λ B, ϕ ] gλ [λ ,ϕAB]+2g2[ϕAB,ϕCD][ϕ , ϕ ] , (2.5) − α AB − αA˙ B AB CD ab ab oabc where tr denotes a trace over the colour indices and Dµ = δ ∂µ + igf Aµc is the covariant derivative in the adjoint representation. The action (2.5) is invariant under the supersymmetry transformations

1 1 α˙ δϕAB = (λαAη B λαBη A)+ εABCDη λ 2 α − α 2 α˙ C D A 1 µν β A AB α˙ CA B δλ = F − σ η +4i(/D ϕ )η 8g[ϕ ,ϕ ]η α −2 µν α β αα˙ B − BC α α˙ δAµ = iλαAσµ ηα˙ iηαAσµ λ , (2.6) − αα˙ A − αα˙ A which can be easily obtained from the ten dimensional transformations (2.2) using (2.3) and (2.4). A manifestly off-shell formulation of =4 supersymmetric Yang–Mills theory N would require the introduction of an infinite set of auxiliary fields as a consequence of the presence of bosonic coordinates associated to the twelve central charges in the =4 superspace, as noticed in [39, 40]. A superfield formulation can be constructed N using analytic superspace, however such an approach is rather involved and cannot be applied to explicit calculations because it is intrinsically on-shell. An on-shell AB AB A superfield description can be given in terms of a superfield W = W (x, θα , θαA˙ ) satisfying [41] the reality condition 1 W = ε W CD , (2.7) AB 2 ABCD 44 Chapter 2. =4 super Yang–Mills theory N

together with the constraint,

AW BC = [AW BC] , (2.8) Dα Dα

where is the super-covariant derivative. The fields entering the action (2.5) are Dα combined into the lowest components of W AB. In the next chapter perturbative calculations in =4 Yang–Mills will be carried N out using a formulation of the model in terms of = 1 superfields. The field content N of the theory can be obtained by taking one =1 vector superfield V and three N =1 chiral superfields ΦI all in the adjoint representation of the gauge group. In N this approach the six real scalars are combined into three complex fields that are the scalar components of the chiral superfields ΦI , three of the Weyl fermions are the spinors of ΦI and the fourth fermion is the gaugino that together with the real vector constitute the vector superfield V . As a result only a SU(3) U(1) subgroup × of the original SU(4) symmetry is manifest in this formulation. More precisely the representations of SU(4) decompose according to 6 3+3, 4 3+1, so that the → → I chiral superfields Φ transform in the 3 of SU(3), the antichiral ΦI† in the 3 and the vector V is a singlet under SU(3). The action in the =1 superfield formulation reads N

1 4 4 gV gV I 1 2 1 α S = tr d x d θ e− ΦI† e Φ + d θ W Wα + h.c. + dr 4g2 4 Z Z √2 2 I J K 2 IJK + ig d θε Φ [Φ , Φ ]+ d θε Φ† [Φ† , Φ† ] , (2.9) 3! IJK I J K Z Z #)

where dr denotes the Dynkin index of the representation to which the ﬁelds belong. To recover the correct dependance on the coupling constant g one must substitute v 2gV in the second term in (2.9). → The action (2.9) is non-polynomial as is always the case in supersymmetric gauge theories in =1 superspace. In particular the scalar potential of (2.5) comes from N the superpotential term

d4x d2θ W(x, θ, θ) + h.c. = Z Z √2 = d4x d2θ i ε f abc(ΦI ΦJ ΦK)(x, θ, θ) + h.c. − 3! IJK a b c Z (Z " # ) 2.1 Formulations of =4 SYM 45 N

By power expanding eV one obtains

S = d4xd4θ V a(x, θ, θ) [ 2P ξ(P + P )2] V (x, θ, θ)+ − T − 1 2 a Z a I a b Ic +Φ†I (x, θ, θ)Φa(x, θ, θ)+ igfabcΦ†I (x, θ, θ)V (x, θ, θ)Φ (x, θ, θ)+ 1 2 e a b c Id g f f Φ† (x, θ, θ)V (x, θ, θ)V (x, θ, θ)Φ (x, θ, θ)+ . . . −2 ab ecd I i 2 gf D DαV a(x, θ, θ) V b(x, θ, θ) D V c(x, θ, θ) + −4 abc α 1 h i 2 g2f ef V a(x, θ, θ) DαV b(x, θ, θ) D V c(x, θ, θ) D V d(x, θ, θ) + −8 ab ecd α √2 h i + . . . gf abc ε ΦI (x, θ, θ)ΦJ (x, θ, θ)ΦK (x, θ, θ)δ(θ)+ − 3! IJK a b c IJK a + ε Φ†Ia(x, θ, θ)Φ†Jb(x, θ, θ)Φ†Kc(x, θ, θ)δ(θ) + C′a(x, θ, θ)C (x, θ, θ)+

a i a a C′a(x, θ, θ)C (x, θ, θ) + gfabc C′ (x, θ, θ)+ C′ (x, θ, θ) − 2√2 · b c c 1 2 e a a V (x, θ, θ) C (x, θ, θ)+ C (x, θ, θ) g f f C′ (x, θ, θ)+ C′ (x, θ, θ) · − 8 ab ecd · d V b(x, θ, θ)V c(x, θ, θ) Cd(x, θ, θ)+ C (x, θ, θ) + . . . , (2.10) · which of course can be made polynomial by choosing the Wess–Zumino gauge which implies V 3 = 0. In (2.10) the gauge fixing term as well as the corresponding ghost fields have been included following the prescription of (1.67) of chapter 1 and the expansion has been truncated to the terms that will be relevant in future calculations. Integrating over the θ variables and eliminating the auxiliary fields through the equations of motion results in a different component field formulation, which again has a manifest SU(3) U(1) global symmetry. Such a formulation will be employed × in the next chapter in the discussion of subtleties related to the problem of gauge fixing in this model. Choosing the Wess–Zumino gauge one obtains

4 1 a a µν µ I i a a i a aI S = d x F F D ϕ† D ϕ + λ Dλ/ + ψ Dψ/ + −4 µν − µ I 2 2 I Z a a i√2 a +ig√2f λ ϕb ψcI ψ ϕbI λc f εI ψ ϕbJ ψcK + (2.11) abc I − I − 2 abc JK I IJ a b cK 1 2 a b 2 1 2 a LMK bI cJ d e ε ψ ϕ† ψ g (f ϕ† ϕ ) + g f f ε ε ϕ ϕ ϕ† ϕ† . − K I J − 2 abc I I 2 abc de IJK L M This formulation can be easily seen to be perfectly equivalent to (2.5) apart from the the manifest R-symmetry, that is only SU(3) U(1). × 46 Chapter 2. =4 super Yang–Mills theory N

It is also useful to give a third component field formulation of =4 supersym- N metric Yang–Mills theory in terms of = 2 multiplets. This approach will actually N be used in instanton calculations in chapter 4. The =4 super Yang–Mills multiplet N decomposes in terms of =2 multiplets into a vector plus a hypermultiplet. In this N description the subgroup of the original SU(4) R-symmetry that is realized explicitly is SU(2) SU(2) U(1), where the subscripts and refer to the vector and hy- V × H× V H permultiplet respectively: fields in the two multiplets are charged only with respect to one of the two SU(2) factors. The representations of SU(2) SU(2) U(1) will V × H× be denoted by (r , r )q with the subscript q referring to the U(1) charge and the V H following notation for the component fields will be used

λu (2, 1) , ϕ (1, 1) , A (1, 1) V → ∈ +1 ∈ +2 µ ∈ 0 u˙ ψ (1, 2) 1, quu˙ (2, 2)0 , (2.12) H → ∈ − ∈ where u, u˙ =1, 2. In particular the form of the Yukawa couplings in this formulation will be of relevance in the following calculations. In =2 notation the Yukawa N couplings read

( =2) √2 a bα u S cu˙ b Suu˙ cα˙ N = gf q λ σ ψ + ψ σ λ + LY 2 abc S uu˙ α α˙ u˙ u n a uv b cα˙ bα u˙ c v˙ + ϕ ε λαu˙ λv + εu˙v˙ ψ ψα +

a bα u c v u˙v˙ b cα˙ + ϕ εuvλ λα + ε ψα˙ u˙ ψv˙ , (2.13) o In conclusion of this review of the diverse formulations of = 4 super Yang– N Mills it is worth recalling that a truly off-shell analysis of the theory, in components and in the Wess–Zumino gauge, was given in [42] within the framework of algebraic renormalization. The problems related to the necessity of including an infinite set of auxiliary fields are dealt with using the method of Batalin and Vilkoviski. This approach, though technically rather involved, allows to prove that all the classical symmetries of the theory survive quantization.

2.2 Classical symmetries and conserved currents

The ﬁeld content of =4 supersymmetric Yang–Mills theory described in the pre- N ceding section is unique apart from the choice of the gauge group G. In the Abelian case the theory is free as follows immediately by observing that all the ﬁelds belong 2.2 Symmetries and currents 47

to the adjoint representation of the gauge group, so that all the covariant derivatives reduce to ordinary ones if G is Abelian, and all the potential terms are proportional to the structure constants fabc. In the non Abelian case the theory has a moduli space of vacua parameterized by the vacuum expectation values (vev’s) of the six real scalars, corresponding to a Coulomb phase. The manifold of vacua is determined by the condition of vanishing scalar potential (F -ﬂatness plus D-ﬂatness) which reads

tr [ϕAB,ϕCD][ϕ , ϕ ] =0 tr [ϕi,ϕj][ϕ ,ϕ ] =0 . AB CD ⇐⇒ i j By writing ϕi explicitly in terms of generators of the adjoint representation one obtains

1,6 dim G 2 2 a b 2 ϕiaϕjb tr [T , T ] =0 , (2.14) i,j a=1 X X so that for a rank r gauge group G at most r components ϕia for each ﬁeld can have non vanishing vacuum expectation value. As a result the space of vacua is parameterized by 6r real coordinates. The moduli space is

= R6r/ , (2.15) M Sr where is the group of permutations of r elements, because configurations in which Sr two or more components of ϕi are exchanged are physically indistinguishable. In conclusion the manifold isa6r-dimensional real space with orbifold type singularities. M In a generic point of the Coulomb phase with nonvanishing vev’s for the scalar fields the gauge group G is broken to U(1)r, so that the theory is again Abelian and thus free. On the contrary the origin of the moduli space, where the vev’s of all the scalars vanish, corresponds to a phase that is expected to describe a highly non trivial superconformal field theory. Note that the superconformal phase corresponds to an orbifold singularity of the manifold . In this phase the classical action (2.5) is M actually invariant under a larger group of global transformations than that generated by the super Poincaréalgebra (1.3), namely the =4 superconformal group. N The -extended superconformal algebra in four dimensions is an extension of N i (1.3) which includes additional generators D, Kµ, Sα and Sαi˙ corresponding respectively to dilatations, special conformal transformations and the associated special supersymmetry transformations. The complete algebra further includes the follow- 48 Chapter 2. =4 super Yang–Mills theory N

ing (anti)commutation relations, supplementing (1.3)

[M ,K ]= η K η K µν λ νλ µ − µλ ν [D,P ]= P [D,K ]= K µ − µ µ µ [P ,K ]= 2M +2η D [K ,K ]=0 µ ν − µν µν µ ν ˙ µν µν β i µν α˙ βi [Sαi, M ]=(σ )α Sβi [Sα˙ , M ]=(σ) β˙ S

j S , S =2σµ K δj { αi α˙ } αα˙ µ i i j S ,S =0 S , S ˙ =0 { αi βj} { α˙ β} 1 1 [Qi ,D]= Qi [Q ,D]= Q α 2 α αi˙ 2 αi˙ 1 i 1 i [S ,D]= S [S ,D]= S αi 2 αi α˙ 2 α˙ − (2.16) αi˙ α˙ [Qi ,Kµ]= iσµ S [Q ,Kµ]= iσµ αα˙ S α αα˙ i − αi µ αi˙ µ [Sαi,K ]=0 [S ,K ]=0

α˙ αi˙ [S ,P µ]= iσµ Q [S ,P µ]= iσµ αα˙ Qi αi − αα˙ i α a aj a i ai j [S , T ]= B S [T , S ]= B† S αi − i αj α˙ − j α˙ Qi ,S =2ε δi D i(σµν ) γε M δi 4iε δi A + Bai T { α βj} αβ j − α γβ µν j − αβ j j a j j µν γ˙ j j aj Q , S ˙ =2ε ˙ δ D iε (σ ) ˙ M δ +4iε ˙ δ A + B† T { αi˙ β} α˙ β i − α˙ γ˙ β µν i α˙ β i i a 4 4 [Qi , A]= i −N Qi [Q , A]= i −N Q α − 4 α αi˙ 4 αi˙ N N 4 i 4 i [S , A]= i −N S [S , A]= i −N S . αi 4 αi α˙ − 4 α˙ N N The operator A in these relations is the generator of the U(1) factor of the group U( )=SU( ) U(1) of the automorphisms of the algebra. The =4 case is singular N N × N since it implies

i [Qα, A] = [Qαi˙ , A]=0 .

The chiral rotation operator A must therefore have the same action on all the states in the multiplet. This implies in particular that it must act in the same way on states of opposite helicities 1 and 1 , which is possible only if A is actually zero. − 2 2 2.2 Symmetries and currents 49

For this reason the R-symmetry group in the = 4 case is SU(4) and not U(4) as N would be expected from general arguments. As already noticed in section 2.1 a SU(4) SO(6) global symmetry, that is identi- ∼ ﬁed with the R-symmetry, naturally arises in the compactiﬁcation on T 6 that leads to =4 super Yang–Mills in D = 4 from the ten dimensional =1 theory. In this N N context the reduction of the global symmetry from U(4) to SU(4) is understood as a consequence of the lack of a real six dimensional representation in U(4). More αA B precisely it follows from the form of the Yukawa coupling λ [λα , ϕAB] in the action (2.5). In fact a transformation, under the U(1) factor contained in U(4), of the iα AB 2iα AB type λ e λ for the spinor would require ϕ e− ϕ for the scalars which is → → prohibited by the reality condition (2.3). The Noether currents associated with the superconformal transformations together with those corresponding to chiral SU(4) R-transformations are components of a unique on-shell multiplet [43]. The complete multiplet of currents was given for the Abelian case in [44]. The conserved currents are

←→ µν 1 µν 2 µ νρ 1 αA (µ ν) α˙ = [δ (F − ) 4F − F − + h.c.] λ σ ∂ λ T 2 ρσ − ρ − 2 αα˙ A + δµν (∂ ϕ )(∂ρϕAB) 2(∂µϕ )(∂ν ϕAB) ρ AB − AB 1 (δµν2 ∂µ∂ν )(ϕ ϕAB) − 3 − AB ←→ µ κν µ α˙ µ B 4 µν β B Σ = σ F − σ λ +2iϕ ∂ λ + iσ ∂ (ϕ λ ) αA − κν αα˙ A AB α 3 α ν AB β ←→ 1 α˙ µ B = ϕ ∂µϕCB + λ σµαα˙ λ B δ BλαC σµ λ . (2.17) J A AC αA˙ α − 4 A αα˙ C The remaining components of the supermultiplet are obtained by supersymmetry using the equations of motion and read

2 = (F − ) C µν A µν β A Λˆ = σ F − λ α − α µν β AB = λαAλ B E α AB αA β B AB = λ σ λ +2iϕ F − (2.18) Bµν µνα β µν 1 χˆC = ε (ϕDEλ C + ϕCEλ D) α AB 2 ABDE α α 1 AB = ϕABϕ δA δB ϕEF ϕ . Q CD CD − 12 [C D] EF In (2.17) is the (improved) energy momentum tensor, Σµ are the spin 1 su- Tµν αA 2 persymmetry currents and µB are the SU(4) currents. is a singlet under SU(4) JA Tµν 50 Chapter 2. =4 super Yang–Mills theory N

whereas Σµ transforms in the 4 and µB in the 4 4. The multiplet further includes αA JA × (AB) ij 1 C ˆ A three scalars , and , two fermionic spin 2 componentsχ ˆα AB and Λα and C E Q[AB] one antisymmetric tensor µν . Their transformation under SU(4) is as follows: the B scalars , (AB) and ij are respectively in the 1, 10 and 20, the two fermionsχ ˆC C E Q α AB A [AB] and Λˆ belong to the 4 6 and 4 respectively and µν is in the 6. α × B All of these ﬁelds can be obtained as lowest components of an on-shell superﬁeld ij which is a bilinear in W i 1 ti W AB W(2) ≡ 2 AB δij ij = tr W iW j W W k . W(2) − 6 k In the non Abelian case in (2.17) and (2.18) there is a trace over the colour indices and additional terms are present. Such terms comprise beyond those required for the covariantization of all the derivatives also ‘potential’ terms. For example the complete non Abelian expression for (AB) is E (AB) = tr λαAλ B + g t(AB)+ tr ϕiϕjϕk , (2.19) E α [ijk] (AB)+ AB where t[ijk] is the completely antisymmetrized product of three ti matrices.

2.3 Finiteness of =4 supersymmetric Yang–Mills N theory

The first argument suggesting the possibility that the =4 supersymmetric Yang– N Mills theory could be a finite theory at the quantum level, free of ultraviolet as well as infrared divergences, was given in [19]. In that paper within the study of the general problem of the coupling of scalar (chiral) multiplets to vector multiplets it was observed that a number of three chiral multiplets in a supersymmetric Yang– Mills theory would lead to a Gell-Mann and Low β function vanishing in the one-loop approximation. In fact for a gauge group G=SU(N) the one loop β function is given by g2 β = (3 n)N , −16π2 − where n is the number of chiral multiplets coupled to the vector multiplet in the =1 N language. Successively this result was improved in [45], where the same was shown to hold at the two loop level by an explicit computation in the component field formulation. 2.3 Finiteness of =4 SYM 51 N

Further improvement of this result beyond two loops proves extremely complicated using ordinary techniques based on Feynman rules for the component theory. There exist a computer calculation [46] of the three-loop β function performed by using ordinary Feynman rules which gives a vanishing result as well. Superspace techniques allow to derive this result in a much simpler way as was shown in [23, 47, 48]. The basic ingredient is the observation that a vertex coupling three chiral superfields is finite as a direct consequence of dimensional analysis. By applying the rules of section 1.7 the superficial degree of divergence of a Green function with three external (anti) chiral superfields is negative, D = 2 3 = 1, i.e. − − the diagram is finite. This means that the β function is completely determined by the wave function renormalization constant calculated from the divergent part of the propagator of, say, the chiral superfield. The equation for the β function [48] is

3 ∂Z(1) β(g)= g2 g , 2 ∂g

(1) where Zg is the coefficient of the simple pole term in the Fourier transform of the chiral superfield propagator. Calculations presented in the papers of references [47, 48] show that the propagator of the chiral superfield is finite, in the supersymmetric generalization of the Fermi–Feynman gauge (α=1 in the notation of section 1.6), up to three loops, implying the vanishing of the β function at the same order in perturbation theory. The lack of manifest =2 supersymmetry in the calculations described up to now N prevents from using the non renormalization theorem for =2 in order to extend the N result to every order. A formulation based on unconstrained = 2 superfields (which N is basically a version of harmonic superspace) was developed in [49]; such an approach allows a description of =4 super Yang–Mills with manifest =2 supersymmetry. N N The finiteness of the theory at all orders in perturbation theory is then a consequence of the =2 non renormalization theorem once the absence of divergences is proved N at one loop. Another proof of the finiteness of the theory was given in [50] using a light-cone superspace formulation. The advantage of this formulation relies on the fact that only physical fields are involved so that no auxiliary fields are necessary, however the lack of Lorentz invariance introduces other difficulties and makes the interpretation of the results less transparent. A different approach based on anomaly arguments has been presented in [51]. 52 Chapter 2. =4 super Yang–Mills theory N

The vanishing of the β function is obtained as a consequence the superconformal invariance since (see [52])

β(g) F a F µν = µ , g µν a Tµ

so that invariance under dilatations, implying the vanishing of µ, immediately gives Tµ β(g) = 0. The problem is therefore the proof of superconformal invariance, which is obtained as a consequence of the SU(4) symmetry being preserved at the quantum level. As already remarked a different demonstration of superconformal symmetry was given in [42] using somewhat more general arguments. All of the results here discussed either apply to gauge invariant quantities or are obtained avoiding possible subtleties related to the choice of the gauge. As will be discussed in the next chapter subtleties are actually present in the calculation of non gauge invariant quantities such as Green functions both in the component field formulation and in the description in terms of =1 superfields. N

2.4 S-duality

During the last few years the concept of duality has been the basic ingredient for striking results in the analysis of the non-perturbative behaviour of supersymmetric gauge theories as well as of superstring theories. The fundamental idea of duality is that of a transformation relating two diﬀerent and somehow complementary descriptions of a physical system. The rˆole that duality plays in the study of the non-perturbative properties of a physical model relies on the possibility of constructing a transformation that maps the non-perturbative regime of the system into the perturbative regime of the ‘dual’ system. Examples of this kind of duality have been known for a long time; in particular the duality shown by Kramers and Wannier [53] in the case of the 2-D Ising model and that established between the Sine-Gordon and the Thirring models [54, 55] appear very representative.

The two dimensional Ising model is deﬁned by a set of spin variables σi taking values 1 and living on a two dimensional square lattice. The interaction is restricted ± to nearest neighbours and the partition function reads

Z(K)= exp K σ σ , (2.20)  i j σ (i,j) X{ } X   2.4 S-duality 53

where K is related to the temperature T and the ferromagnetic coupling constant J through K = J/k T (k being Boltzmann’s constant). In (2.20) the sum on σ is B B { } over all spin conﬁgurations and the sum on (i, j) is over all pairs of nearest neighbours. The model was explicitly solved by Onsager and exhibits a single phase transition to a ferromagnetic state at a critical temperature Tc. Kramers and Wannier computed the critical temperature by using duality arguments, before the exact solution of the model was found. The crucial point is that the theory can be reformulated in a diﬀerent (“dual”) way by writing the partition function as a sum over the elementary plaquettes of a dual lattice whose sites are the centers of the plaquettes of the original one, with a new coupling K∗. The two descriptions are then equivalent, namely

Z∗(K∗)= Z(K), if the couplings are related by 1 sinh 2K∗ = . sinh 2K Notice that this implies that the high temperature (or weak coupling) regime, K 1, ≪ of the original description is mapped by the duality transformation to the low temperature (or strong coupling) regime, K∗ 1, of the dual system. The computation of ≫ the critical temperature is based on the hypothesis that the system has a single transition point. The critical point must then be the self-dual point of the transformation,

K = K∗, which gives sinh[2J/kBTc]=1. A second interesting example of duality occurs in two dimensional relativistic field theory and relates the Sine-Gordon and the Thirring models. The Sine-Gordon model is defined by the 2-D action 1 α S = d2x ∂ φ∂µφ + (cos βφ 1) , (2.21) SG 2 µ β2 − Z where φ is a scalar field and the parameter β plays the rôle of a coupling constant as can be established by power expanding the potential term. The spectrum of the theory contains ‘meson’ excitations of mass Mm = √α and solitonic states with 8√α mass Ms = β2 . Therefore the solitonic degrees of freedom have large mass in the weak coupling regime. It can be shown that the Sine-Gordon model is completely equivalent to the Thirring model which describes a system of interacting fermions. The action of the latter is g S = d2x ψiγ ∂µψ + mψψ ψγ ψψγµψ . (2.22) T µ − 2 µ Z h i ST can be mapped into the action (2.21) through bosonization [54, 55]. More precisely it can be proved that the duality transformation maps the soliton of the Sine-Gordon 54 Chapter 2. =4 super Yang–Mills theory N

model into the elementary fermion of the Thirring model and the meson states of (2.21) into fermion–anti-fermion bound states. The relation between the coupling constants is

β2 1 = g , 4π 1+ π

so that in this case as well duality establishes a strong/weak coupling correspondence. The crucial rôle that duality plays in the context of supersymmetric gauge theories relies on this fundamental feature. The kind of duality that is relevant in four dimensional quantum field theories is a generalization of the electric-magnetic duality introduced for the first time by Dirac in [56]. The general features of electric-magnetic duality are reviewed in appendix B, where the case of the Georgi–Glashow (or Yang– Mills–Higgs) model is described. The application to the =4 super Yang–Mills N theory is discussed in this section. There are various reviews on electric-magnetic duality, see in particular [58, 59, 60].

2.4.1 Montonen–Olive and SL(2,Z) duality

Most of the analysis of this section will be carried out for a SU(2) gauge group, but the results can be generalized to larger groups. The classical spectrum of non Abelian gauge theories can contain non-perturbative states associated with solitonic solutions of the equations of motion. The corresponding ﬁeld conﬁgurations are characterized by a topological charge that is interpreted as a magnetic charge, so that these states describe monopoles and in general dyons (states with both electric and magnetic charge) [61]. As discussed in appendix B the Georgi–Glashow model with Lagrangian

1 1 λ 2 = F a F aµν + DµΦaD Φ ΦaΦ v2 , L −4 µν 2 µ a − 4 a − possesses, in the (BPS) limit of vanishing potential, a ‘classical’ spectrum, contain-

ing one scalar massive Higgs ﬁeld, one massless ‘photon’, massive W ± bosons and

monopoles M ±, that is summarized in the following table 2.4 S-duality 55

Mass (Qe, Qm) Spin Higgs 0 (0,0) 0 Photon 0 (0,0) 1

W ± ve ( e,0) 1 ± M ± vg (0, g) 0 ±

where Qe and Qm are the electric and magnetic charge operators deﬁned as 1 Q = d3x D ΦaBi m v i a Z 1 Q = d3x D ΦaEi , (2.23) e v i a Z i i with Ea and Ba the non Abelian electric and magnetic ﬁelds respectively. All of the states saturate the Bogomol’nyi bound [62]

m v Q2 + Q2 , (2.24) ≥ e m p where Q = 4πnm and Q = n e, with n , n Z. Of course quantum corrections m e e e e m ∈ might in principle modify this situation, however the above picture is expected to be a good approximation in the weak coupling limit, e 1. In this limit ≪ 4π m = ev v , m = gv = v v . W ≪ M e ≫ Montonen and Olive have suggested in [37] that a duality transformation exchanging “electric” states associated to W ± bosons and “magnetic” states associated to monopoles M ± and at the same time electric and magnetic charges could be an exact symmetry of the full quantum theory in the BPS limit. The exchange of electric and magnetic charges implies, because of the Dirac quantization condition (see appendix B),

4π e g = , −→ e so that the proposed symmetry is intrinsically non-perturbative. The conjecture of [37] is based on the classical spectrum and on the following arguments. First of all the mass formula (2.24) is invariant under duality as can be easily checked. Moreover a semiclassical analysis [63] based on the so called moduli 56 Chapter 2. =4 super Yang–Mills theory N

space approximation (to be discussed in the ﬁnal subsection) shows that the interaction between two monopoles vanishes, whereas there is an attractive interaction between monopoles and anti-monopoles. The duality symmetry would require that

the same should be true for the W ± bosons. It can be proved at the semi-classical level that this is indeed the case because the repulsive interaction between equal charge W bosons is exactly cancelled by an identical contribution due to the exchange of scalars, that are massless in the BPS limit. There are some basic problems with the Montonen–Olive proposal, namely

Quantum corrections are expected to spoil the results of the semi-classical anal- • ysis. In particular a non-vanishing potential V (Φ) could be generated.

The elementary W ± bosons have spin one while the monopoles that should be • related to them by the duality have spin zero, making an exact matching of the quantum numbers impossible.

The rˆole of the dyons has been neglected in the previous analysis. • It will be shown that all of these problems can be solved in supersymmetric theories. In particular the =4 supersymmetric Yang–Mills theory is supposed to realize N exactly a generalization of the Montonen–Olive duality called S-duality. This generalization emerges when the eﬀects of a non-vanishing vacuum angle are taken into account. In the presence of a θ-term (see the end of section B.2) the Lagrangian of the Georgi–Glashow model can be written

1 1 = Im [τ (F µν + i F µν)(F + i F )] DµΦD Φ , (2.25) L −32π ∗ µν ∗ µν − 2 µ

θ 4π with τ = 2π + i e2 . The electric and magnetic charges of the dyon states saturating eθ 4π the Bogomol’nyi bound are Qe = nee + nm 2π and Qm = nm e respectively. Periodicity in θ requires invariance under

τ τ + b , b Z , (2.26) −→ ∈ which amounts to a shift in the electric charge n n bn , n n , or in matrix e → e − m m → m form n 1 b n e e . (2.27) nm −→ 0 1 nm 2.4 S-duality 57

Furthermore at θ = 0 the duality of Montonen and Olive becomes 1 τ , (2.28) −→ −τ that is equivalent to the exchange of electric and magnetic charges, e g = 4π , → e g e. In Montonen–Olive duality this is supplemented by the exchange of the → − corresponding quantum numbers, Q Q , implying n n , n n . The e ↔ m e → m m → − e latter transformation can be written in matrix form as n 0 1 n e e . (2.29) nm −→ 1 0 nm − It is thus natural to generalize the duality symmetry to the set of transformations generated by (2.26) and (2.28) with arbitrary θ. The resulting group of transformations is SL(2,Z) acting projectively on the parameter τ

aτ + b τ , a,b,c,d Z , ad bc =1 . (2.30) −→ cτ + d ∈ −

The corresponding action on the quantum numbers ne and nm, generated by (2.27) and (2.29), is

n a b n e − e . (2.31) nm −→ c d nm − One can then easily check that the mass formula, written in terms of τ as

2 2 1 1 Re τ ne m 4πv (ne, nm) − 2 (2.32) ≥ Im τ Re τ τ nm − | | is invariant.

2.4.2 S-duality in =4 supersymmetric Yang–Mills theory N The general properties of the =4 supersymmetric Yang–Mills theory previously N discussed allow to give an intuitive explanation of why it is considered the original example of a theory possessing exact S-duality. In the Coulomb phase for each scalar with non-vanishing vev the same construction leading to the ’t Hooft–Polyakov monopole (see the appendix) can be carried on. As a result the classical spectrum contains monopoles and dyons as well. Unlike the case of the Georgi–Glashow model the ﬁniteness of the theory makes the semiclassical analysis of the spectrum sensible even at the quantum level. Moreover the theory has the correct number of fermions in 58 Chapter 2. =4 super Yang–Mills theory N

the adjoint representation to give spin one dyon/monopole conﬁgurations through the mechanism sketched in section B.3. More precisely, in the case of a SU(2) gauge group i i being discussed here, creation and annihilation operators a † and a , with i = 1, 2, ± ± associated with the fermionic zero modes can be deﬁned so that the spectrum of zero-energy states, in the charge-one monopole sector, consists of the following states [64]

where Ω is the magnetic charge-one Clifford vacuum. The multiplet generated in | i this way contains sixteen states: eight bosons (six spin 0 scalars and two spin 1 ± vectors) and eight spin 1 fermions. As a result the spin quantum numbers of the ± 2 monopole multiplet exactly match those of the multiplet of elementary fields. Furthermore in extended supersymmetric theories the Bogomol’nyi bound comes as a consequence of the supersymmetry algebra, so that it holds in the full quantum theory. In supersymmetric models a vanishing scalar potential in correspondence of non trivial scalar field configurations is perfectly natural because of the presence of flat directions. In the case of =4 super Yang–Mills this scalar field configuration N corresponds to a generic point of the Coulomb phase. The relation of the BPS limit with the supersymmetry algebra is better understood in the =2 case. The =2 supersymmetry algebra possesses two central N N charges which are related to the magnetic and electric charge operators of the theory [65]. Using a Majorana notation, the =2 algebra contains the anticommutator N

Qi , Qj = δijγµ P + δ U ij +(γ ) V ij , i, j =1, 2 , { α β} αβ µ αβ 5 αβ 2.4 S-duality 59

where the central charges have been denoted by U ij and V ij. By expressing U ij and V ij in terms of the ﬁelds of the theory Olive and Witten have proved the relations

ij ij ij ij U = ε vQe , V = ε vQm , where Qe and Qm are the electric and magnetic charge operators respectively, that generalize those in equation (2.23). In the previous equation v2 = tr(S2) + tr(P 2) , h i h i where S and P denote the complex Higgs ﬁelds of the model. As a consequence of this relation the Bogomol’nyi bound follows directly from the supersymmetry algebra because of the positive deﬁniteness of the operator Q, Q in the rest frame. { } In the =4 case the situation is more involved because there are twelve central N charges. Correspondingly there are six electric and six magnetic charge operators associated to the six scalars, ϕi, of the =4 multiplet N i 1 ~ i Qe = d~σ tr Eϕ v S2 · Z ∞ i 1 ~ i Qm = d~σ tr Bϕ , (2.33) v S2 · Z ∞ where v2 = 6 tr(ϕi)2 . The Bogomol’nyi bound reads i=1h i P m2 v2 (Qi )2 +(Qi )2 ≥ e m and can be saturated only by states annihilated by exactly one half of the supercharges. The BPS states which saturate the limit correspond to short multiplets of the =4 superalgebra and for this reason are protected in the full quantum theory. N The mass of the BPS states is completely determined by their electric and magnetic quantum numbers and is exact because supersymmetry is preserved at the quantum level. All the BPS states in the spectrum can be displayed on six two-dimensional lattices i i (one for each couple of charges Qe and Qm) and lie at points of coordinates eθ 4π = Q + iQ = n e + n + in . Q e m e m 2π m e

Supersymmetry implies that the BPS states in the spectrum can only decay into a couple of BPS states, so that a generic state of mass m is unstable if and only if there exist BPS states of masses m1 and m2 such that

m m + m . ≥ 1 2 60 Chapter 2. =4 super Yang–Mills theory N

Figure 2.1: Lattice of BPS states

It immediately follows that stable BPS states are all and only those which have

integer quantum numbers ne and nm that are coprime. These are the states marked by empty circles in ﬁgure 2.1, while the full circles denote unstable states. Thanks to the fact that the BPS states are protected by supersymmetry in the =4 theory a semi-classical analysis of the spectrum is sensible at the quantum level N as well. Some arguments supporting the conjectured S-duality of the =4 super N Yang–Mills theory based on the moduli space approximation are brieﬂy reviewed in the next subsection.

2.4.3 Semiclassical quantization

In appendix B it is discussed how to construct a classical solution of the equations of motion describing a monopole in the Georgi–Glashow model, the aim of this section is to show how the semiclassical quantization of the monopole solution can be used to study the spectrum of the model [63, 66]. The notion of collective coordinates will be introduced starting with the ’t Hooft–Polyakov monopole and then the semiclassical quantization will be applied to the =4 super Yang–Mills case. A readable review N of these topics is provided by [67]. a In the BPS limit the monopole solution satisfies the Bogomol’nyi equation Bi = a DiΦ (i =1, 2, 3, a =1,..., dim G). Given a field configuration solving this equation in general one can find a multi-parameter family of solutions with the same energy. 2.4 S-duality 61

The parameters labeling the degenerate solutions are called moduli or collective coordinates of the monopole and the corresponding space moduli space. For the ’t Hooft–Polyakov monopole described in appendix B three collective coordinates are associated with the translational invariance of the Bogomol’nyi equation. cl a cl a Given a solution (Ai (~x), Φ (~x)) the ﬁeld conﬁguration

cl a ~ cl a ~ Ai (~x + X) , Φ (~x + X) , (2.34) where X~ is a constant vector, still satisfies the equation and has the same energy (and magnetic charge). Thus the three coordinates X~ are moduli. A fourth collective coordinates is related to gauge transformations. Gauge invariance implies that the functional space of physical field configurations is

= / , C A G where =(Aa, Φa) and is the group of “small” gauge transformations, i.e. gauge A i G transformations that do not reduce to the identity at spatial infinity. “Large”, or “global”, gauge transformations which do not approach the identity at infinity are true symmetries of the theory and so field configurations related by large gauge transformations are inequivalent. The BPS solution for the ’t Hooft–Polyakov monopole is symmetric under the diagonal SO(3) subgroup of SO(3) SU(2) , where SO(3) is R× G R the group of spatial rotations and SU(2)G the group of large gauge transformations, but it is not invariant under SU(2)G. As a result by acting with the U(1) subgroup of SU(2)G that is not broken in the Higgs vacuum one generates a new solution with the same energy, so that the fourth collective coordinate is associated with the U(1) group of global ‘electromagnetic’ transformations. By letting X~ depend on time one cl a ~ cl a ~ describes a moving monopole, (Ai (~x+X(t)), Φ (~x+X(t))). Since the Hamiltonian for the system in the temporal gauge A0 = 0 is 1 1 H = T + V = d3x , A˙ aA˙ i + Φ˙ aΦ˙ + d3x , BaBi + D ΦaDiΦ 2 i a a 2 i a i a Z Z h i the time dependence of X~ generates a non vanishing electric field so that the kinetic energy increases while the potential energy remains constant. This is a general feature of a motion in the moduli space. One can then consider a general deformation of a

BPS monopole in the A0 = 0 gauge

a a δAi (~x, t) , δΦ (~x, t) . (2.35) 62 Chapter 2. =4 super Yang–Mills theory N

To keep the potential energy ﬁxed (2.35) must satisfy the linearized Bogomol’nyi equation

ε DjδAk D δΦ+ e[δA , Φcl]=0 (2.36) ijk − i i and the Gauss law constraint

i cl DiδA + e[Φ , δΦ]=0 . (2.37)

The general solution can be written in terms of a single function of time χ(t)

cl δAi = Di χ(t)Φ δΦ = 0 (2.38) δA = D χ(t)Φcl χ˙(t)Φcl . 0 0 − a a ~ If χ is taken to be constant and (δAi , δΦ ) does not contain a translation δ~x = ~x+X, the deformation reduces to a large gauge transformation, so that χ is identiﬁed with the fourth collective coordinate. Forχ ˙ = 0 a non-vanishing kinetic energy is gen- 6 erated. Moreover χ is periodic since it parameterizes a compact U(1) (it is a sub-

group of SU(2)G), therefore the moduli space with coordinates (X,~ χ) is topologically = R3 S1. M1 × The previous construction of the moduli space can be generalized to a charge k monopole solution. A rigorous derivation of multimonopole solutions is quite involved [68], the resulting moduli space is 4k-dimensional and roughly speaking the Mk 4k collective coordinates correspond to the locations of the k monopoles and their dyon degrees of freedom. On physical grounds the existence of multimonopole BPS configurations is related to the fact that the repulsion between monopoles of the same charge can be compensated by an attractive interaction mediated by the massless Higgs field, thus allowing the construction of static configurations. a a For a charge k monopole deformations (δαAi , δαΦ ), with α =1,...,k, satisfying the linearized Bogomol’nyi equations and the Gauss law constraint define tangent vectors to . Given the tangent vectors the standard metric on can be con- Mk Mk structed as

= d3x tr δ A δ Ai + δ Φδ Φ . (2.39) Gαβ − α i β α β Z This metric is actually induced by the form of the action for the gauge theory in C cl cl as can be seen by writing the monopole solution as Ai (~x, zα), Φ (~x, zα), where zα’s 2.4 S-duality 63

denote the moduli. The tangent vectors (δαAi, δαΦ) are calculated by differentiating with respect to the zα. In general one has to include a gauge transformation to ensure that (δαAi, δαΦ) are tangent to the moduli space of physical fields, i.e. that the fields cl α cl α A′ = A + δ A δz , Φ′ = Φ + δ Φδz belong to the space . The general form of i i α i α C the tangent vectors is ∂A δ A = i D ǫ α i ∂zα − i α ∂Φ δ Φ = e[Φcl, ǫ ] , α ∂zα − α

β where ǫα(~x, z ) is a suitable gauge parameter that is fixed by requiring that the linearized Bogomol’nyi equation (2.36) and the Gauss law constraint (2.37) be satisfied. The action of the theory on the BPS configuration is then obtained after substituting

A (~x, t) Acl(~x, zα(t)) i −→ i A z˙α(t)ǫ (2.40) 0 −→ α Φ(~x, t) Φcl(~x, zα(t)) −→ and reads 1 4πv S = d3xdt tr F clF 0i = dt z˙αz˙β k . (2.41) −2 0i cl Gαβ − e Z Z The action (2.41) is the starting point for the semiclassical quantization which is achieved by studying the corresponding quantum mechanics. In the k = 1 case the moduli are zα =(X,~ χ) and the action becomes

1 4πv ˙ 2 4π 4πv S = dt X~ + χ˙ 2 . (2.42) 2 e ve2 − e Z The resulting wave functions are plane waves

iP~ X~ ineχ ψ = e · e , with n Z, and one can compute the electric charge, Q = ie∂ , so that the e ∈ e − χ quantum mechanical system possesses an inﬁnite tower of states with Qe = nee. For the mass of these states one gets n2ve3 4πv m = e + v[Q2 + Q2 ] , 8π e ≈ e m showing that the Bogomol’nyi bound is saturated. 64 Chapter 2. =4 super Yang–Mills theory N

The preceding arguments can be extended to the case of the =4 super Yang– N Mills theory to ﬁnd stronger evidence for the exact S-duality of the model. In order to be able to use the previous considerations the simplest case will be considered, namely the gauge group will be taken to be G=SU(2), broken by the vacuum conﬁguration

ϕ2 = ϕ3 = . . . = ϕ6 =0 h i h i h i ϕ1 = Φ , trΦ2 = v2 . h i h i The perturbative states in the bosonic sector of the theory, labeled by quantum

numbers (nm, ne) consist of a massless photon multiplet with charge (0, 0) and massive W boson multiplets with charge (0, 1). The tower of stable BPS states predicted ± ± by S-duality have charges (l,k) (with l and k coprimes) and can be studied within the framework of semiclassical quantization since they can be put in correspondence with certain geometric structures on . Then the non-renormalization properties Mk of the theory make the semiclassical result reliable in the full quantum theory. In the Georgi–Glashow model is parameterized by 4k collective coordinates Mk which are bosonic zero modes. In the =4 supersymmetric Yang–Mills theory there N are fermionic zero modes as well in the monopole background and the index theorem of [71] shows that the number of such zero modes in the charge k sector is precisely 4k. As a consequence the moduli space has fermionic coordinates too and the low energy ansatz to be substituted into the Bogomol’nyi equation is given by (2.40) supplemented by

λ(~x, t) ψ(t)λcl(~x, z (t)) (2.43) ∼ α for the Weyl spinors in the monopole background. In equation (2.43) the ansatz for the spinors λ(~x, t) has been schematically written as the product of a c-number cl λ (~x, zα) satisfying a zero mode equation in the monopole background and a time dependent fermionic collective coordinate ψ(t). The zero modes can be shown to constitute a short multiplet preserving half of the supersymmetries just like is expected for BPS states. The exact ansatz for the ﬁelds is rather complicated and was discussed in detail in [69]. A technically involved computation allows to derive the quantum mechanical action to be employed in the semiclassical analysis. The action reads

1 α 1 α β 4πv S = dt z˙αz˙β + iψ γ0D ψβ + R ψ ψγ ψ ψδ k , (2.44) 2 Gαβ t 6 αβγδ − e2 Z 2.4 S-duality 65

α ˙ α α β γ where a Majorana notation is used. In (2.44) Dtψ = ψ +Γβγz˙ ψ is the covariant derivative, with Γα the Christoffel symbols of the metric , and R is the βγ Gαβ αβγδ corresponding Riemann tensor [69]. It can be shown that thanks to the fact that the monopole moduli space is a hyper-Kähler manifold, S in equation (2.44) is invariant under the action of eight real supercharges. The quantization of the supersymmetric quantum mechanics (2.44) was performed in [70], where it was shown that the states can be put in one to one correspondence with differential forms on and the Mk Hamiltonian of the system H can be expressed in terms of the exterior derivative acting on forms. This construction leads to a basis of sixteen independent forms that are eigenstates of H in the nm = k = 1 sector. These states constitute a BPS multiplet as can be verified by a computation of the corresponding quantum numbers. The complete wave functions contain a plane wave factor from the bosonic sector of

iP~ X~ ineχ eθ the form e · e so that again one ﬁnds a tower of states with Qe = ne + 2π . These states all saturate the Bogomol’nyi bound and therefore are exactly those predicted by S-duality.

For nm = k> 1 the problem is much more complicated. The moduli space is

= R3 (S1 ˜ 0)/Z , Mk × × Mk k where ˜ 0 is a 4(k 1)-dimensional hyper-K¨ahler manifold. The states are tensor Mk − products of forms on R3 S1 with forms on ˜ 0, s = ω, n α 2. The forms × Mk | i | ei⊗| i α can be proved to be unique and much in the same way as in the k = 1 case there | i are sixteen forms ω, ne for all k that give rise to a BPS multiplet. The resulting | i eθ states carry electric charge Qe = nee + nm 2π and saturate the Bogomol’nyi bound. The unique form α for the case k = 2 has been explicitly determined by Sen in [72]. | i Arguments suggesting a possible generalization to k > 2 were given in [73].

2 There are actually further subtleties related to the Zk identiﬁcation. Chapter 3

Perturbative analysis of =4 supersymmetric Yang–MillsN theory

As discussed in the previous chapter, the =4 supersymmetric Yang–Mills theory N has been proved to be finite at the perturbative level up to three loops and strong arguments have been proposed in order to extend this result to all orders and even non perturbatively. For this reason the theory has long been considered to be rather trivial. However recent developments, within the context of the correspondence with type IIB superstring theory on anti de Sitter space, that will be studied in the final chapter, have shown that it is actually an extremely interesting superconformal field theory, possessing very peculiar properties. Even in perturbation theory a careful analysis allows to point out various problems mostly related to the choice of gauge. Both in the component field formulation and using =1 superfields the gauge fixing procedure appears very subtle. As will be N discussed in detail in both cases one finds divergences in off-shell Green functions. In the component formulation the propagators of the elementary fields are ultraviolet divergent in the Wess–Zumino (WZ) gauge and these infinities are exactly cancelled when the contributions of the gauge-dependent fields, that are put to zero in the WZ gauge, are taken into account. The choice of the WZ gauge, that is almost unavoidable in explicit computations, introduces divergences that require a wave function renormalization. This is a general result that applies to theories with less supersymmetry as well, but appears particularly dramatic in the =4 case be- N 66 67

cause the theory is finite, so that the divergences cannot be reabsorbed through a redefinition of the bare parameters. Different problems emerge in the formulation of the theory in terms of =1 super- N fields. Almost all the calculations showing the vanishing of the quantum corrections to two- and three-points functions that are presented in the literature were performed in the supersymmetric generalization of the Fermi–Feynman gauge (α=1 in the notation of section 1.6). With a different choice of gauge two and three point functions develop infrared singularities of difficult interpretation, leading to the result that the choice α=1 is somehow privileged. This conclusion was proposed for the first time in [74] and then it was discussed in the case of the =4 theory in [75]; however no N possible explanation was proposed. Unlike those of the elementary fields the correlation functions of gauge invariant composite operators, that play a crucial rôle in the correspondence with AdS type IIB supergravity/superstring theory, should not suffer from problems related to the gauge fixing. Also, it will be discussed, in the superfield formulation, the effect of the introduction of a mass term for the chiral superfields, which breaks supersymmetry from =4 down to =1. It will be shown that this deformation of the model does not N N modify the ultraviolet properties of the original theory. This result was first proposed in [76], where it was proved that the inclusion of mass terms for the (anti) chiral superfields does not generate divergent corrections to the effective action. It will be argued that this statement can be reinforced, showing that, at least at one loop, no new divergences, not even corresponding to wave function renormalizations, appear as a consequence of the addition of the mass terms. This result supports the claim put forward in [77, 78], where the ‘mass deformed’ =4 theory was proposed as a N supersymmetry-preserving regularization scheme for a class of =1 theories. It will N also be argued that the same approach should work in the case of =2 theories. N All of this problems are examined in detail in this chapter. Section 3.1 deals with the difficulties introduced by the choice of the Wess–Zumino gauge when the component formalism is used. The subsequent sections report calculations performed in the =1 superfield formalism of two-, three- and four-point functions. The material N presented in this chapter is an original and systematic re-analysis of the perturbative properties of =4 supersymmetric Yang–Mills theory, the treatment follows closely N [79]. 68 Chapter 3. Perturbative analysis of =4 SYM theory N

3.1 Perturbation theory in components: problems with the Wess–Zumino gauge

Throughout this chapter calculations will be carried on in Euclidean space-time and unless otherwise stated the gauge group will be taken to be SU(N). To discuss perturbation theory in components in this section the formulation of equation (2.11), with a SU(3) U(1) subgroup of the SU(4) R-symmetry group manifest, will be em- × ployed. To correctly deal with the lower components of the vector superfield V , that are put to zero in the Wess–Zumino gauge, it is however useful not to eliminate the auxiliary fields F and D through their equations of motion, but rather keep them in the action: in the computation of Green functions the corresponding x-space propagators are simply δ-functions. The complete expression for the vector superfield V

was given in equation (1.24) and contains beyond the physical fields, Aµ and λ, and the auxiliary field, D, the ‘gauge-dependent’ fields C, χ and S. Note that the scalar field C and the spinor χ have “wrong” physical dimension, resulting in non standard

free propagators, as will be shown. The non Abelian ﬁeld strength superﬁeld Wα is

∞ 1 V V (k) W = DDe− D e = W , (3.1) α −4 α α Xk=1 where 1 W (1) = DDD V α −4 α (3.2) 1 W (2) = DD[V,D V ] α 8 α

(k) and the terms Wα , with k 3, contain k factors of V and vanish in the WZ gauge. ≥ The Euclidean action in =1 superﬁelds can be written N

(E) 1 4 4 I 1 1 (1)α (1) 1 (1) (1)α ˙ S = tr d xd θ ΦI† Φ W Wα δ(θ)+ W α˙ W δ(θ)+ dr − − 4g2 4 4 Z 1 2 2 I 1 (1)α (2) 1 (2)α (2) D VD V g[Φ† V ]Φ W W + W W δ(θ)+ −8α − I − 8g2 α 16g2 α 1 (1) (2)α ˙ 1 (2) (2)α ˙ 1 2 I + W W + W W δ(θ) g [V, [V, Φ† ]]Φ + . . . , (3.3) 8g2 α˙ 16g2 α˙ − 2 I where the gauge ﬁxing term has been included, whereas no ghost term is displayed since it will not be relevant for the computations to be discussed in this section. In equation (3.3) the dots denote terms of higher order in V , which do not contribute 3.1 Perturbation theory in components 69

to the Green functions that will be considered and that will be suppressed from now on. In the following calculations the Fermi–Feynman gauge, α=1, will be used. With this choice one obtains 1

1 4 4 2 I 1 (1)α (2) 1 (2)α (2) S = tr d xd θ V V ΦI† Φ W Wα + W Wα δ(θ)+ dr − − 8g2 16g2 Z h i 1 (1) (2)α ˙ 1 (2) (2)α ˙ I 1 2 I + W W + W W δ(θ) g[Φ† V ]Φ + g [V, [V, Φ† ]]Φ , 8g2 α˙ 16g2 α˙ − − I 2 I where from the deﬁnition (3.2) it follows

1 i W (1) = iλ + δβD δβ2C (σµσν) β (∂ A ∂ A ) θ + α − α α − 2 α − 2 α µ ν − ν µ β µ α˙ +θθσαα˙ ∂µλ . (3.4)

(2) The explicit form of Wα will not be necessary for the moment (see however equation (3.18)). Expansion of the action in components using the complete expression of V gives

S = S0 + Sint .

I The free action S comes from the terms V 2V and Φ† Φ and reads 0 − I

4 a µ I a µ I a I 2 a S0 = d x (∂µϕI†)(∂ ϕa)+ ψI σ (∂µψa)+ FI †Fa + Sa† S + Z 1 1nh 1 1 1 i 1 Ca2D Da2C Ca22C + χa2λ + χa2λ + λa2χ + −2 a − 2 a − 2 a 2 a 2 a 2 a 1 a 1 1 1 + λ 2χ + χa2σµ(∂ χ )+ χa2σµ(∂ χ )+ (∂ Aa)(∂ A ) . (3.5) 2 a 2 µ a 2 µ a 2 µ ν ν aµ

The interaction part Sint contains an inﬁnite number of terms. The propagators of the fermion ψ and of the scalar ϕ belonging to the =1 chiral multiplet will now be N computed at one loop. The terms that are relevant for these calculations come from 2 the expansion of Φ†V Φ and Φ†V Φ in the superﬁeld action. The latter generates tadpole type diagrams in the propagator ϕ†ϕ , but not in ψψ . The interaction h i h i 1From now on the superscript E will be suppressed. Euclidean signature is understood unless otherwise stated. 70 Chapter 3. Perturbative analysis of =4 SYM theory N

part of the action to be considered is

4 1 a b cI 1 a b cI 1 a b µ cI S = d x igf φ †D φ + φ †(2C )φ (∂ φ †)C (∂ φ )+ int abc 2 I 2 I − 2 µ I Z i a b µ cI µ a b cI i a b cI a b cI + φ †A (∂ φ ) (∂ φ †)A φ + φ †S †F F †S φ + 2 I µ − I µ √2 I − I a b cI i a b cI a b cI i a b cI a b cI F †C F + ψ λ φ φ †λ ψ + F †χ ψ ψ χ F + − I √2 I − I √2 I − I i a b µ cI a µ b cI 1 b a µ cI b a µ cI φ †(∂µχ )σ ψ + ψ σ (∂µχ )φ + C ψ σ (∂µψ ) C (∂µψ )σ ψ + −√2 I 2 I − I i a µ cI b √2 IJK a b c a b c aI bJ cK ψ σ ψ A gf ε φ †φ †F † φ †ψ ψ + ε φ φ F + −2 I µ − 3! abc I J K − I J K IJK 2 g a φaI ψbJ ψcK f f e CbψdI σµψ Ac + − − 2 abe cd − I µ i c d b aI i dI µ a dI µ a b c a b c +(χ ψ )(χ ψ )+ ∂ ψ σ ψ ψ σ ∂ ψ C C + ϕ † C D + I 2 µ I − µ I I 1 c b c i µ c b c i µ c b c + 2C χ λ + σ ∂ χ χ λ + σ ∂ χ + S †S 2 − 2 µ − 2 µ − 1 bc cµ dI i b cµ a dI a dI 1 b c a dI A A ϕ + C A ϕ †(∂ ϕ ) (∂ ϕ †)ϕ + C C ϕ †(2ϕ )+ −2 µ 2 I µ − µ I 4 I a dI i b µ c a dI a dI +(2ϕ †)ϕ + χ σ χ ϕ †(∂ ϕ ) (∂ ϕ †)ϕ . (3.6) I 2 I µ − µ I The quadratic part of the action, S0, can be written in a more compact form introducing the notation χa(x) a a C (x)  χ (x)  a(x)= , a(x)= , (3.7) B a F a D (x) !  λ (x)   a   λ (x)    so that  

4 a µ I a µ I a I S0 = d x (∂µϕI†)(∂ ϕa)+ ψI σ (∂µψa)+ FI †Fa + Z nh i a 1 a µ T a T a + S†2S A A + M + N , (3.8) a − 2 µ a Ba B Fa F where 0 2σµ∂ 2 0 µ − 1 22 2 1 2σµ∂ 0 0 2 M = − − , N =  µ −  . (3.9) 2 2 0 2 2 0 0 0 − −  0 2 0 0   −    3.1 Perturbation theory in components 71

Inverting the kinetic matrices M and N one gets the free propagators. From (3.9) it follows 1 0 0 0 −2  1  1 0 0 0 0 −2 1 2 1   M − =  1  , N − =  1 σµ∂  , (3.10)  µ  2 1  2 0 0 2   −   −     µ   1 σ ∂µ   0 2 2 0   −    so that the free propagators are J, b I, a b I b I δaδJ bI ϕ †(x)ϕ (y) = δ(x y) = ∆ (x y) −→ h J a ifree − 2 − aJ −

b a b b δa b S †(x)S (y) = δ(x y)=2∆ (x y) −→ h a ifree 2 − a −

b a δb Cb(x)D (y) = a δ(x y) = ∆b (x y) −→ h a ifree 2 − a −

b a Db(x)D (y) = δbδ(x y) −→ h a ifree − a −

J, b I, a b I I b F †(x)F (y) = δ δ δ(x y) −→ h J a ifree J a −

µ, b ν, a δ δb Ab (x)A (y) = µν a δ(x y) = ∆b (x y) −→ h µ aν ifree − 2 − aµν −

α,b β,a εαβδb χbα(x)λβ(y) = a δ(x y)= Rbαβ(x y) −→ h a ifree − 2 − a −

˙ α˙ β˙ b α,˙ b β, a β˙ ε δ bα˙ β˙ χbα˙ (x)λ (y) = a δ(x y)= R (x y) −→ h a ifree − 2 − a −

b µ α,˙ b α, a bα˙ δaσ ∂µ λ (x)λα(y) = αα˙ δ(x y)= −→ h a ifree 2 − b =S (x y) aαα˙ − 72 Chapter 3. Perturbative analysis of =4 SYM theory N

b J µ α,I,b˙ α, J, a bα˙ δaδI σ ∂µ ψ (x)ψαJ (y) = αα˙ δ(x y)= −→ h I a ifree 2 − bJ =S (x y) aIαα˙ − In conclusion beyond the ordinary propagators for the physical ﬁelds, ϕ, ψ, λ and

Aµ, and those for the auxiliary ﬁelds, F and D, one further obtains the propagators

S†S , CD , χλ and χλ . The latter are not present in the Wess–Zumino gauge. h i h i h i h i

3.1.1 One loop corrections to the propagator of the fermions belonging to the chiral multiplet

The one-loop correction to the propagator of the fermions ψI in the chiral multiplet is the simplest to compute. In the WZ gauge there are three contributions at the one loop level, that will be shown to lead to a logarithmically divergent result. From the action (3.5), (3.6), with C=χ=S=0 one obtains the following three diagrams

AµAν

a ψ (x) ψI (y) AaI (x; y) J b −→ bJ ψψ ψψ

a ψ (x) ψI (y) BaI (x; y) J b −→ bJ

ϕϕ†

λλ

a ψ (x) ψI (y) CaI (x; y) J b −→ bJ

ϕ†ϕ

The three contributions depicted above can be easily evaluated and give the results

1 αβaI˙ AααaI˙ (x; y) = g2f d f l d4x d4x ∆me(x x ) S (x x )σν bJ −2 ef mn 1 2 νµ 2 − 1 lL − 2 βγ˙ · Z γγnL˙ n βαfK˙ h S (x x )σµ S (x y) , (3.11) · dK 2 − 1 γβ˙ bJ 1 − io 3.1 Perturbation theory in components 73

1 BααaI˙ (x; y) = g2εLMN εPQ f f f d4x d4x ∆ld (x x ) bJ −2 R de lmn 1 2 PL 2 − 1 · Z αβaI˙ βαnR˙ n S (x x )Sem (x x )S (x y) , (3.12) · fN − 2 ββMQ˙ 2 − 1 bJ 1 − h io CααaI˙ (x; y) = g2f f f n d4x d4x ∆lK (x x ) bJ − de lm 1 2 fL 2 − 1 · Z αβaI˙ n βαdL˙ S (x x )Sem(x x )S (x y) . (3.13) · nK − 2 ββ˙ 2 − 1 bJ 1 − Taking the Fourier transformh one obtains io

αβ˙ βα˙ 4 ˜ααaI˙ i 2 2 a I ˜ λ ˜ d k 1 AbJ (p)= g N(N 1)δb δJ S (p)σββ˙ pλS (p) 4 2 2 , 2 − " (2π) k + p k p # Z 2 − 2 B˜ααaI˙ (p)= C˜ααaI˙ (p)= A˜ααaI˙ (p) , bJ − bJ bJ where αα˙ σαα˙ pµ S˜ (p)= i µ . − p2 In conclusion the one-loop correction to the fermion propagator in the Wess–Zumino gauge turns out to be logarithmically divergent

ααaI˙ 1 loop,WZ ααaI˙ ααaI˙ ααaI˙ ψψ − = A˜ (p)+ B˜ (p)+ C˜ (p)= h bJ iFT bJ bJ bJ αβ˙ βα˙ 4 i 2 2 a I ˜ λ ˜ d k 1 = g N(N 1)δb δJ S (p)σββ˙ pλS (p) 4 2 2 . (3.14) 2 − " (2π) k + p k p # Z 2 − 2 Equation (3.14) shows that a logarithmically divergent wav e function renormalization is required in the WZ gauge. This divergence will now be proved to be a gauge artifact due to the choice of the WZ gauge. In fact if the ﬁelds C, χ and S are included two further contributions must be taken into account, which correspond to the diagrams

λχ

a ψ (x) ψI (y) DaI (x; y) J b −→ bJ

ϕ†ϕ λχ

a ψ (x) ψI (y) EaI (x; y) J b −→ bJ

ϕ†ϕ 74 Chapter 3. Perturbative analysis of =4 SYM theory N

The contribution of these two diagrams exactly cancels the divergence in equation (3.14) giving a net vanishing one-loop result for the propagator. In fact the computation of D and E gives

1 DααaI˙ (x; y)= g2f f f n d4x d4x ∆lK (x x ) bJ −2 de lm 1 2 fL 2 − 1 · Z αγaI˙ βαdL˙ S (x x )σµ ∂(2)Rγne˙ (x x ) S (x y) , (3.15) · mK − 2 γγ˙ µ β˙ 2 − 1 bJ 1 − h io ααaI˙ ααaI˙ EbJ (x; y)= DbJ (x; y) .

In momentum space one has

αβ˙ βα˙ 4 ˜ ααaI˙ i 2 2 a I ˜ λ ˜ d k 1 DbJ (p)= g N(N 1)δb δJ S (p)σββ˙ pλS (p) 4 2 2 , −4 − " (2π) k + p k p # Z 2 − 2 ˜ααaI˙ ˜ ααaI˙ EbJ (p)= DbJ (p) .

In conclusion summing up all the terms gives

ααaI˙ 1 loop ψψ − = h bJ iFT ˜ααaI˙ ˜ααaI˙ ˜ααaI˙ ˜ ααaI˙ ˜ααaI˙ = AbJ (p)+ BbJ (p)+ CbJ (p)+ DbJ (p)+ EbJ (p)=0 , (3.16)

so that the one-loop correction to the ψψ propagator, if the Wess–Zumino gauge is h i not exploited to perform the calculation, is zero, as expected in =4 super Yang– N Mills theory. Notice that the insertion of tadpoles such as

in a diagram gives a vanishing result because all the propagators are diagonal in colour space, so that the tadpole contains a factor δ f abc 0. The same is true also ab ≡ for diagrams in =1 superspace that will be discussed in subsequent sections. N 3.1.2 One loop corrections to the propagator of the scalars belonging to the chiral multiplet

The calculation of the propagator ϕ†ϕ is more complicated because more diagrams h i 2 are involved. In particular tadpole type graphs, coming from the term Φ†V Φ in the 3.1 Perturbation theory in components 75

action, are present. However the result is analogous to what was found for the ψψ h i propagator: in the WZ gauge the one-loop correction is logarithmically divergent, requiring a wave function renormalization, but when the contributions neglected in the WZ gauge are considered the total one-loop result is vanishing.

The diagrams contributing to ϕ†ϕ at the one-loop level in the Wess–Zumino h i gauge are the following

AµAν

a I aI ϕ †(x) ϕ (y) A (x; y) J b −→ bJ

ϕ†ϕ

λλ

a I aI ϕ †(x) ϕ (y) B (x; y) J b −→ bJ ψψ

ψψ

a I aI ϕ †(x) ϕ (y) C (x; y) J b −→ bJ ψψ 76 Chapter 3. Perturbative analysis of =4 SYM theory N

AµAν

a I aI ϕ †(x) ϕ (y) D (x; y) J b −→ bJ

ϕ†ϕ

a I aI ϕ †(x) ϕ (y) E (x; y) J b −→ bJ DD

F F †

a I aI ϕ †(x) ϕ (y) F (x; y) J b −→ bJ

ϕϕ†

The last three diagrams are all tadpole corrections since the free propagators for the auxiliary ﬁelds F and D are δ-functions, so that the corresponding lines shrink to a point. Each of these diagrams is quadratically divergent. There is also a quadratically divergent contribution coming from the two diagrams B(x, y) and C(x, y) and the sum of all these terms exactly vanishes. The sum of the previous diagrams gives a ﬁnal result that is logarithmically divergent and in momentum space is schematically of the form

4 aI 1 loop,WZ 2 I a 2 d k 1 (ϕ†ϕ)bJ FT− g δJ δb p . (3.17) h i ∼ (2π)4 k + p 2 k p 2 Z 2 − 2

Just like in the case of ψψ this divergence corresponds to a wave function renor- h i malization for the ﬁeld ϕ(x).

Taking into account the contribution of the ﬁelds C, χ and S there are eight more diagrams to be calculated 3.1 Perturbation theory in components 77

χλ

a I aI ϕ †(x) ϕ (y) G (x; y) J b −→ bJ ψψ

λχ

a I aI ϕ †(x) ϕ (y) H (x; y) J b −→ bJ ψψ

a I aI ϕ †(x) ϕ (y) L (x; y) J b −→ bJ ψψ

F †F

a I aI ϕ †(x) ϕ (y) M (x; y) J b −→ bJ

SS†

λχ

a I aI ϕ †(x) ϕ (y) N (x; y) J b −→ bJ λχ

a I aI ϕ †(x) ϕ (y) P (x; y) J b −→ bJ

a I aI ϕ †(x) ϕ (y) Q (x; y) J b −→ bJ 78 Chapter 3. Perturbative analysis of =4 SYM theory N

SS†

a I aI ϕ †(x) ϕ (y) R (x; y) J b −→ bJ

Single diagrams contain quadratically divergent terms that cancel in the sum leaving a total contribution that is again logarithmically divergent. This non-vanishing correction is exactly what is needed to cancel the divergence in (3.17). As a result

the complete one-loop correction to the ϕ†ϕ propagator is zero, whereas in the h i Wess–Zumino gauge a wave function renormalization is required.

Notice that the situation induced by the choice of the WZ gauge is a feature common to every supersymmetric gauge theory, since the contribution of the gauge- dependent fields C, χ and S is in general logarithmically divergent. However in theories with less supersymmetry a logarithmic wave function renormalization is usually necessary in any case, so that this effect is completely irrelevant. On the contrary it becomes important in the =4 Yang–Mills theory and in general in finite theo- N ries. Of course the divergences encountered here are gauge artifacts and disappear in gauge invariant correlation functions. Examples of computations of correlators of gauge invariant operators, in which the choice of the WZ gauge does not lead to this kind of problems, will be considered within the discussion of the correspondence with AdS type IIB superstring theory in the final chapter. Note also that the divergences encountered in the propagators are zero on-shell.

The calculations described here seem to suggest the possibility of constructing improved Feynman rules in which the effect of the gauge dependent fields, neglected in the WZ gauge, is dealt with by a suitable redefinition of the free propagators. However this program proves extremely complicated when the propagators of fields in the vector multiplet or three- and more-point functions are considered. The calculation of these correlation functions requires, already at the one-loop level, many other (1) (2) terms to be included in the action Sint. In particular terms coming from W W and W (2)W (2) must be considered. The calculation of W (2) without the simplifications 3.2 Propagators in the =1 formulation 79 N introduced by the Wess–Zumino gauge is rather lengthy and gives

a(2) i a b c 1 µ αb˙ c i µ αb˙ c 1 b c W = f iC λ σ χ ∂ C σ χ A + S †χ + α −2 bc − α − 2 αα˙ µ − 2 αα˙ µ √ α 2 b c β β b c i µ ν β b c c + S †S δ + δ C D (σ σ ) C (∂ A ∂ A )+ α α − 2 α µ ν − ν µ i 1 c (σµσν) β(Ab ∂ Cc + Ab ∂ Cc) δ βCb2Cc δ βχbλ + −2 α ν µ µ ν − 2 α − α 1 + (σµσν ) β∂ Cb∂ Cc χbβλc λbβχc iσµ χbα˙ ∂ χcβ+ 2 α µ ν − α − α − αα˙ µ 1 cα˙ +iεγβσµ ∂ χbχc + (σµσν) βAb Ac θ + Cbσµ ∂ λ + γβ˙ µ α 2 α ν µ β αα˙ µ 1 1h iχb Dc (σµσν) βχb (∂ Ac ∂ Ac ) Ab (σµσν) β∂ χc + − α − 4 α β µ ν − ν µ − 2 ν α µ β 1 bµ c i b µ ν β c i µ αb˙ c b c + ∂µA χ ∂ν C (σ σ )α ∂µχ σ ∂µχ S + √2S λ + 2 α − 2 β − √2 αα˙ α cα˙ iAb σµ λ θθ (3.18) − µ αα˙ i

Substituting (3.18) into Sint results in a number of relevant interaction terms of the order of 100, making this formulation totally impractical in explicit calculations. In conclusion perturbation theory in components requires almost unavoidably the WZ gauge, but the latter introduces divergences in gauge dependent quantities. In the following sections the superfield formalism, that allows to avoid these problems, will be employed. However as will be shown new difficulties related to the gauge fixing emerge.

3.2 Perturbation theory in =1 superspace: prop- N agators

The difficulties encountered in the previous section in the calculation of gauge dependent quantities, because of the divergences introduced by the Wess–Zumino gauge, can be overcome using the =1 superfield formulation of the model. N In chapter 2 it has already been pointed out how the =1 superfield formalism N has proved to be a powerful tool in the proof of the finiteness of the theory up to three loops. In this approach there is no particular difficulty in working without fixing the WZ gauge. If one does not choose to work in the WZ gauge the action is non polynomial, however a finite number of terms is relevant at each order in perturbation 80 Chapter 3. Perturbative analysis of =4 SYM theory N

theory, so that only at very high order the Wess–Zumino gauge introduces significant simplifications. The aim of this section is to show that there are other subtleties related to the gauge fixing, even if one does not work in the WZ gauge. The formulation of the theory in =1 superspace that will be employed is a ‘mass N deformed’ version of that given by equation (2.10). In other words the action that will be considered is that of equation (2.10) plus mass terms for the (anti) chiral superfields

1 S = d4xd4θ mδ ΦI (x, θ, θ)ΦJa(x, θ, θ)δ(θ)+ m − 2 IJ a Z 1 IJ a + m∗δ Φ† (x, θ, θ)Φ† (x, θ, θ)δ(θ) . (3.19) 2 I Ja The addition of this term to the action breaks supersymmetry down to =1. In [76] it N was argued, by means of dimensional arguments, that this term should not affect the ultraviolet properties of the =4 theory. More precisely the statement of [76] is that N no divergences appear in gauge invariant quantities, so that no divergent contribution to the quantum effective action is generated perturbatively. As a result the model obtained in this way would be an example of a finite =1 theory. The inverse N construction, in which one deforms a =1 model to =4 super Yang–Mills plus a N N mass term, has been proposed in [77, 78] as a regularization procedure preserving supersymmetry. The calculations presented in this section will show that in the presence of the term (3.19) no divergence is generated, at the one-loop level, in the two-, three- and four-point Green functions that are computed. The results presented suggest the possibility of reinforcing the conclusions of [76]; namely the =4 theory N augmented with (3.19) appears to be finite in the sense that no divergences appear in the total n-point irreducible Green functions, at least at one loop. In particular no wave function renormalization is required. Notice that this result is what is actually necessary for the consistency of the approach advocated in [77, 78]. The complete action is 2

S = d4xd4θ V a(x, θ, θ) [ 2P ξ(P + P )2] V (x, θ, θ)+ − T − 1 2 a Z a I 1 I Ja +Φ† (x, θ, θ)Φ (x, θ, θ) mδ Φ (x, θ, θ)Φ (x, θ, θ)δ(θ)+ I a − 2 IJ a 1 IJ a a b mδ Φ† (x, θ, θ)Φ† (x, θ, θ)δ(θ)+ igf Φ† (x, θ, θ)V (x, θ, θ) (3.20) −2 I Ja abc I · 2In the following the mass parameter m will be taken to be real for simplicity of notation. 3.2 Propagators in the =1 formulation 81 N

Ic 1 2 e a b c Id Φ (x, θ, θ) g f f Φ† (x, θ, θ)V (x, θ, θ)V (x, θ, θ)Φ (x, θ, θ)+ · − 2 ab ecd I i 2 gf D DαV a(x, θ, θ) V b(x, θ, θ) D V c(x, θ, θ) + −4 abc α 1 h i 2 g2f ef V a(x, θ, θ) DαV b(x, θ, θ) D V c(x, θ, θ) D V d(x, θ, θ) + −8 ab ecd α √2 h i + . . . gf abc ε ΦI (x, θ, θ)ΦJ (x, θ, θ)ΦK (x, θ, θ)δ(θ)+ − 3! IJK a b c IJK a + ε Φ†Ia(x, θ, θ)Φ†Jb(x, θ, θ)Φ†Kc(x, θ, θ)δ(θ) + C′a(x, θ, θ)C (x, θ, θ)+

a i a a C′a(x, θ, θ)C (x, θ, θ) + gfabc C′ (x, θ, θ)+ C′ (x, θ, θ) − 2√2 · b c c 1 2 e a a V (x, θ, θ) C (x, θ, θ)+ C (x, θ, θ) g f f C′ (x, θ, θ)+ C′ (x, θ, θ) · − 8 ab ecd · d V b(x, θ, θ)V c(x, θ, θ) Cd(x, θ, θ)+ C (x, θ, θ) + . . . , (3.21) · where the dots stand for terms that are not relevant for the considerations of this chapter. Notice that in the action (3.21) a gauge ﬁxing term corresponding to a family 1 of gauges parameterized by α = ξ has been introduced. It will now be shown, by explicitly computing the propagators of both the chiral and the vector superﬁelds, that the supersymmetric generalization of the Fermi–Feynman gauge, corresponding to α = 1, is somehow privileged (see also [75, 80]), since any other choice of the parameter α leads to infrared divergences in Green functions.

3.2.1 Propagator of the chiral superﬁeld

The propagator of the chiral superﬁeld is the simplest Green function to compute. The calculation will be reported in detail in order to illustrate the superﬁeld technique. It follows from the action (2.10) that there are three diagrams contributing to the propagator ΦΦ† at the one loop level. The one-particle irreducible parts of these h i diagrams will be directly evaluated in momentum space using the super Feynman rules of section 1.6. The convention employed is that all momenta are taken to be incoming. The diagrams are the following 82 Chapter 3. Perturbative analysis of =4 SYM theory N

a I Φ †(z) Φ (z′) A(z; z′) J b −→

ΦΦ†

Φ†Φ

a I Φ †(z) Φ (z′) B(z; z′) J b −→

Φ†Φ

a I Φ †(z) Φ (z′) C(z; z′) J b −→

where z =(x, θ, θ) and the notation for the internal propagators is

I, a J, b aI b I a 1 Φ (z)Φ †(z′) = δ δ δ (z z′) −→ h J ifree J b 2 + m2 8 −

a b a a δb V (z)V (z′) = [1+(α 1)(P + P )]δ (z z′) −→ h b ifree − 2 − 1 2 8 −

P and P in the VV propagator are the projectors (1.63). Moreover in the presence 1 2 h i of mass terms for Φ and Φ† there are free propagators ΦΦ and Φ†Φ† , that will h i h i enter the calculation of the vector superﬁeld propagator at one loop

I, a J, b 2 aI J IJ a m D Φ (z)Φ (z′) = δ δ δ (z z′) −→ h b ifree − b 4 2(2 + m2) 8 −

I, a J, b 2 a a m D Φ †(z)Φ† (z′) = δ δ δ (z z′) −→ h I bJ ifree − IJ b 4 2(2 + m2) 8 −

Using these free propagators and the rules of section 1.6, with the vertices read 3.2 Propagators in the =1 formulation 83 N from the action (3.21), the three contributions can be evaluated as follows

4 I cf d k 4 4 a 1 2 δJ δ δ(1, 2) A˜(p) = d θ d θ igf Φ †(p, θ , θ ) D (2π)4 1 2 acd I 1 1 −4 1 (p k)2 + m2 · Z − ←− de 1 2 2 2 2 2 δ δ(1, 2) bJ D1 1+ γ(D1D1 + D1D1) 2 igfefbΦ ( p, θ2, θ2) , · −4 − k − ) where γ =(α 1) and the compact notation δ(1, 2) = δ (θ θ )δ (θ θ ) has been − 2 1 − 2 2 1 − 2 introduced. The computation uses the properties of the δ-function, which imply for example [23, 4]

←− ←− D1αδ(1, 2) = δ(1, 2)D2α , D1˙αδ(1, 2) = δ(1, 2)D2˙α , − ←− ←− − ←− ←− 2 2 2 2 D1˙αD1αδ(1, 2) = δ(1, 2)D2˙αD2α , D1D1δ(1, 2) = δ(1, 2)D2D2 , and integrations by parts on the Grassmannian variables in order to remove the D and D derivatives from one δ, so that one θ integration can be performed immediately. In this way one obtains an expression that is local in θ as is expected from the =1 N non-renormalization theorem. For A˜(p) one obtains

2 4 2 2 a I 1 d k 4 4 1 A˜(p) = g N(N 1)δ δ d θ d θ Φ† (p, 1) − − b J 4 (2π)4 1 2 k2[(p k)2 + m2] aI · Z − n 2 2 2 ΦbJ ( p, 2) D D2δ(1, 2) 1+ γ(D2D + D D2)δ(1, 2) = A˜ (p)+ A˜ (p) . · − 1 1 1 1 1 1 1 2 h i h io The ﬁrst term is trivially calculated using [23, 4]

4 2 2 d θ D D δ(θ θ′) δ(θ θ′)=16 − − Z h i 4 m n d θ D D δ(θ θ′) δ(θ θ′)=0 if (m, n) = (2, 2) (3.22) − − 6 Z and gives

4 2 2 a I d k 4 1 bJ A˜ (p)= g N(N 1)δ δ d θ Φ† (p,θ, θ)Φ ( p,θ, θ) . 1 − − b J (2π)4 k2[(p k)2 + m2] aI − Z − n (3.23)o

In the computation of the second term one must use the (anti) commutators of covariant derivatives, see appendix A, which in particular imply [23, 4]

2 2 2 2 D D2D = 162D D2D D2 = 162D2 . (3.24) 84 Chapter 3. Perturbative analysis of =4 SYM theory N

Then integration by parts analogously gives

4 2 2 2 2 a I d k 4 [(p k) + p ] bJ A˜ (p)= γg N(N 1)δ δ d θ − Φ† (p,θ, θ)Φ ( p,θ, θ) . 2 − b J (2π)4 k4[(p k)2 + m2] aI − Z − n (3.25)o In conclusion from the ﬁrst diagram one obtains two contributions, the ﬁrst proportional to γ and the second independent of it. The second diagram gives one single γ-independent contribution. The Feynman rules give

4 d k 4 4 √2 I a B˜(p)= d θ d θ gε f Φ† (p, 1) (2π)4 1 2 − 3! KL acd I · Z ( ! ce K ←− d L 1 δ δ δ(1, 2) 1 2 δ δ δ(1, 2) D 2 M D f N · −4 1 (p k)2 + m2 −4 2 k2 + m2 · − " #

√2 MN f bJ εJ fbe Φ ( p, 2) . · − 3! ! − )

Proceeding exactly as for A˜1 one obtains

4 bJ d k Φ† (p,θ, θ)Φ ( p,θ, θ) B˜(p)= g2N(N 2 1)δaδI d4θ aI − . (3.26) − b J (2π)4 (k2 + m2)[(p k)2 + m2] Z ( − ) From the last diagram one gets

4 2 d k 4 g d a C˜(p)= d θ f f Φ †(p, 1) (2π)4 1 − 2 ac deb I · Z ce 2 2 δ δ(1, 1) 1+ γ(D D2 + D2D ) ΦbJ ( p, 1) . · − 1 1 1 1 k2 − The only non-vanishing contribution comes from the term in which the projection operators act on the δ-function, since δ (θ θ) = 0, so that 4 − 4 2 2 b I d k 4 1 a J C˜(p)= γg N(N 1)δ δ d θ Φ †(p,θ, θ)Φ ( p,θ, θ) , (3.27) − − a J (2π)4 k4 I b − Z n o from which one sees that the C contribution is absent in the gauge α = 1, i.e. γ = 0. Putting the various corrections together gives the following result. The sum of

the terms A˜1(p) and B˜(p), which does not depend on γ, is

4 2 2 b I d k 4 a J A˜ (p)+ B˜(p) = g N(N 1)δ δ d θ Φ †(p,θ, θ)Φ ( p,θ, θ) 1 − a J (2π)4 I b − · Z 1 h 1 i . · k2 [(p k)2 + m2] − (k2 + m2) [(p k)2 + m2] − − 3.2 Propagators in the =1 formulation 85 N

This is ﬁnite and exactly vanishes for m = 0, i.e. in the limit in which the =4 N theory is recovered.

The sum of the terms A˜2(p) and C˜(p) is proportional to γ and reads

4 2 2 b I d k 4 a J A˜ (p)+ C˜(p) = γg N(N 1)δ δ d θ Φ †(p,θ, θ)Φ ( p,θ, θ) 2 − a J (2π)4 I b − · Z (p k)2 1 h p2 i − + . · k4 [(p k)2 + m2] − k4 k4 [(p k)2 + m2] − − Both of the terms in the last integral are infrared divergent. The ﬁrst one is zero in the limit m 0, while the second one gives an infrared divergence that is still → present in the =4 theory, i.e. with m = 0. More precisely putting N 2 2 b I a J A˜ (p)+ C˜(p)= γg N(N 1)δ δ Φ †(p,θ, θ)Φ ( p,θ, θ) [I (p)+ I (p)] , 2 − a J I b − 1 2 h i one has d4k (p k)2 1 I (p)= − = 1 (2π)4 k4 [(p k)2 + m2] − k4 Z − d4k m2 = = (2π)4 k4 [(p k)2 + m2] Z − 1 d4k 1 =2m2 dζ ζ = 4 2 2 2 3 0 (2π) [k +(p ζ + m )(1 ζ)] Z Z − 2m2 1 1 = dζ ζ = 2(4π)2 (p2ζ + m2)(1 ζ) Z0 − m2 1 m2 p2 + m2 = log ǫ + log −(4π)2 (p2 + m2) p2(p2 + m2) m2 where a standard Feynman parameterization has been used. Analogously

d4k p2 I (p)= = 2 (2π)4 k4 [(p k)2 + m2] Z − 1 d4k 1 =2p2 dζ ζ = 4 2 2 2 3 0 (2π) [k +(p ζ + m )(1 ζ)] Z Z − p2 1 1 = dζ ζ = (4π)2 (p2ζ + m2)(1 ζ) Z0 − 1 p2 m2 p2 + m2 = log ǫ + log . −(4π)2 (p2 + m2) (p2 + m2) m2 In the above expressions an infrared regulator ǫ has been introduced. Notice that the total correction is exactly zero on-shell, i.e. for p2=m2. 86 Chapter 3. Perturbative analysis of =4 SYM theory N

To summarize the results, the propagator of the chiral superﬁelds of the =4 N super Yang–Mills theory in =1 superspace is logarithmically infrared-divergent for N any choice of the gauge parameter α = 1. This divergence corresponds to a wave 6 function renormalization for the superﬁelds ΦI and vanishes on-shell. In the Fermi– Feynman gauge α=1 the one-loop correction exactly vanishes.

3.2.2 Propagator of the vector superﬁeld

The one-loop calculation of the propagator of the vector superfield is much more complicated, because many more diagrams are involved producing a large number of contributions. The final result is however completely analogous: in the Fermi– Feynman gauge the one-loop correction is zero, whereas infrared divergences arise for α = 1. In the presence of a mass term for the (anti) chiral superfields no new 6 divergences are generated.

Form a calculational viewpoint the new feature with respect to the ΦΦ† case h i is that there are also diagrams involving the ghosts. There are two multiplets of

ghosts, described by the chiral superﬁelds C and C′. The free propagators for these superﬁelds will be denoted by

a b a b a 1 C ′(z)C (z′) = δ δ (z z′) −→ h ifree b 2 8 −

a b a b a 1 C (z)C ′(z′) = δ δ (z z′) −→ h ifree b 2 8 −

The ghosts are treated exactly like ordinary chiral superﬁelds with the only diﬀer-

ence that there is a minus sign associated with the loops, because C and C′ are anticommuting ﬁelds [23].

The corrections to the VV propagator at the one-loop level correspond to the h i diagrams 3.2 Propagators in the =1 formulation 87 N

ΦΦ

a V (z) V (z′) A(z; z′) b −→

Φ†Φ†

ΦΦ†

a V (z) V (z′) B(z; z′) b −→

Φ†Φ

ΦΦ†

a V (z) V (z′) C(z; z′) b −→

CC′

a V (z) V (z′) D (z; z′) b −→ 1

C′C

a V (z) V (z′) D (z; z′) b −→ 2

CC′

C′C

a V (z) V (z′) D (z; z′) b −→ 3

CC′ 88 Chapter 3. Perturbative analysis of =4 SYM theory N

C′C

a V (z) V (z′) E (z; z′) b −→ 1

CC′

a V (z) V (z′) E (z; z′) b −→ 2

a V (z) V (z′) F (z; z′) b −→ VV

a V (z) V (z′) G(z; z′) b −→

The contributions A to E are rather straightforward to evaluate much in the same

way as the diagrams entering the ΦΦ† propagator. The last two graphs are more h i involved because the free propagator for the V superﬁeld is more complicated for generic values of the parameter α. Moreover the cubic and quartic vertices

i 2 α a b c gfabc D (D V ) V (DαV ) −16√2 1 h i 2 g2f ef V a DαV b D V c D V d , −128 ab ecd α h i

lead to various terms corresponding to the ways the covariant derivatives can act on 3.2 Propagators in the =1 formulation 89 N the V lines. The V 3 vertex in particular

2 b

3 c

1 a gives rise to six different terms. Schematically the calculation goes as follows. A˜(p) is a contribution that appears because of the addition of the mass terms (3.19); it is not present in the =4 the- N ory, i.e. when m=0. It is useful to discuss separately the corrections coming from diagrams A˜, B˜ and C˜ and those obtained from D˜ i, E˜i, F˜ and G˜, since the latter correspond to the one-loop contribution to the vector superfield propagator in the =1 supersymmetric Yang–Mills theory. N The first diagram, A˜, is logarithmically divergent and reads

d4k m2 V a(p,θ, θ)V b( p,θ, θ) A˜(p)=3g2N(N 2 1)δ d4θ − . (3.28) − ab (2π)4 (k2 + m2)[(p k)2 + m2] Z ( − )

For the diagram B˜ application of the Feynman rules gives rise to three diﬀerent contributions, one quadratically divergent and two logarithmically divergent

d4k 1 B˜(p)=3g2N(N 2 1)δ d4θ − ab (2π)4 (k2 + m2)[(p k)2 + m2] · Z − i α˙ k2V a(p,θ, θ)V b( p,θ, θ) p σµ V a(p,θ, θ) (D Dα)V b( p,θ, θ) + · − − 4 µ αα˙ − h i 1 2 + V a(p,θ, θ) (D D2)V b( p,θ, θ) . (3.29) 4 − h i The tadpole diagram C˜ gives a quadratically divergent result of the form

d4k 1 C˜ = 3g2N(N 2 1)δab d4θ V a(p,θ, θ)V b( p,θ, θ) . (3.30) − − (2π)4 (k2 + m2) − Z Summing and subtracting m2 in the numerator of the ﬁrst term in (3.29) and putting the three corrections A˜, B˜ and C˜ together gives a net result that is only logarithmi- 90 Chapter 3. Perturbative analysis of =4 SYM theory N

cally divergent d4k 1 A˜(p)+ B˜(p)+ C˜(p) = 3g2N(N 2 1)δ d4θ − − ab (2π)4 (k2 + m2)[(p k)2 + m2] · Z − i α˙ p σµ V a(p,θ, θ) (D Dα)V b( p,θ, θ) + · 4 µ αα˙ − h i 1 2 + V a(p,θ, θ) (D D2)V b( p,θ, θ) . (3.31) 16 − h i Notice that in particular the logarithmically divergent contribution proportional to m2 exactly cancels out. This is crucial because this correction would correspond to a mass renormalization for the vector superﬁeld that is known to be excluded in any gauge theory as well as in supersymmetric theories in general.

The diagrams D˜ i and E˜i are completely analogous to the previous ones, with the only diﬀerence that the mass does not appear in the denominators and there is a minus sign associated with the loops. Their sum is logarithmically divergent and takes the form 1 D˜ (p)+ D˜ (p)+ D˜ (p)+ E˜ (p)+ E˜ (p)= g2N(N 2 1)δ 1 2 3 1 2 8 − ab · 4 d k 1 α˙ d4θ ip σµ V a(p,θ, θ) (D Dα)V b( p,θ, θ) + · (2π)4 k2(p k)2 µ αα˙ − Z − n h i 1 2 + V a(p,θ, θ) (D D2)V b( p,θ, θ) . (3.32) 8 − h i All of these corrections do not depend on the gauge parameter γ=α 1 and must − be summed to those coming from the last two diagrams. F˜ exactly vanishes for any α, so that only E˜ needs to be considered. This diagram produces in principle 72 corrections because the Feynman rules give rise to 18 terms (distributing the covariant derivatives associated with the two vertices), each of which splits into 4, since the free propagator itself contains two terms. Many of these contributions can be easily shown to vanish using the properties of the covariant derivatives. In particular one uses

2 2 2 D DαD D =0 ,

3 which follows from D = 0 and use of the (anti) commutation relations for the D’s. It is useful to separate in the non-vanishing part terms proportional to γ and γ2, from the γ-independent terms. The latter combine to give a logarithmically divergent correction that together with that of equation (3.32) cancel the correction 3.2 Propagators in the =1 formulation 91 N

(3.31) coming from the sum A˜ + B˜ + C˜. Actually if m = 0 this sum is ﬁnite and 6 exactly vanishes at m=0. As a result the only non-vanishing corrections to the vector superﬁeld propagator at the one loop level come from terms proportional to γ and to γ2 in E˜(p). The former contain an infrared divergent part of the form

µ 4 ν d k σαα˙ σ ˙ pµpν α˙ β˙ J (1)(p)= c γg2δa d4θ ββ V (p,θ, θ) DαD D DβV b( p,θ, θ) , 1 b (2π)4 k4(p k)2 a − Z − n h io but is ultraviolet finite. Furthermore there is a correction, finite both in the ultraviolet and in the infrared regions, proportional to γ2, that reads d4k (σνσµσλ) (p k) k p J (2)(p) = c γ2g2δa d4θ αα˙ − µ ν λ 2 b (2π)4 k4(p k)4 · Z 2 − α˙ (D Dα)V (p,θ, θ) (D2D )V b( p,θ, θ) . · a − nh i h io (1) (2) In the previous expressions for J and J c1 and c2 are non-vanishing numerical constants proportional to N(N 2 1). − In conclusion in the =4 theory, i.e. when m=0, the one-loop correction to the N vector superfield propagator, just like that of the chiral superfield, is infrared singular unless the Fermi–Feynman gauge, α=1, is chosen, in which case it vanishes. Like in the case of the chiral superfield propagator the non-vanishing γ-dependent corrections correspond to a wave function renormalization for the superfield V and are zero on- shell, i.e. for p2=0. The proof of the finiteness of the theory in the presence of the mass terms (3.19) given in [76] is based on naive power counting, which gives the superficial degree of divergence, d , of a diagram in =1 superspace [24] s N d =2 E C , s − − where E is the number of external (anti) chiral lines and C the number of ΦΦ or

Φ†Φ† propagators. In [76] it is also argued that for corrections to the eﬀective action involving only V superﬁelds the requirement of gauge invariance reduces the degree of divergence to

d = C . s − However for the purposes of [77, 78] it appears crucial that no divergences, not even corresponding to a wave function renormalization, are present in total n-point functions for any n. The computation of the two-point functions in this section has shown 92 Chapter 3. Perturbative analysis of =4 SYM theory N

that this actually the case at one loop. An argument for the generalization of this result to Green functions with an arbitrary number of external V lines will now be briefly discussed. First note that diagrams involving only vector and ghost superfields are not modified by the inclusion of (3.19), so that one must only consider graphs containing internal chiral lines. The ultraviolet properties of diagrams that are only logarithmically divergent in the original =4 theory are not modified by the N presence of the mass in the propagators. Thus the contributions to be analyzed are the quadratically divergent ones, that can acquire subleading logarithmic singulari-

ties, plus eventually new diagrams involving ΦΦ and Φ†Φ† propagators. The relevant quadratic divergences come from tadpole diagrams

V˜ an V˜ bn

V˜ a2 V˜ b2

V˜ a1 V˜ b1

The only new diagram containing ΦΦ and Φ†Φ† propagators that must be considered is

V˜ an V˜ bn

V˜ a2 V˜ b2

V˜ a1 V˜ b1

Since all the vertices entering these graphs come from the expansion of the term V V tr e− Φ†e Φ in the action, the only covariant derivatives that are present come from the functional derivatives and must act on the internal (anti) chiral lines to give a non-vanishing result. As a consequence the loop integral in the above diagrams is exactly the same as for the C˜ and A˜ corrections to the VV propagator. Hence 3.2 Propagators in the =1 formulation 93 N summing to the previous diagrams the contribution of

V˜ an V˜ bn

V˜ a2 V˜ b2

V˜ a1 V˜ b1 leads to a net logarithmically divergent correction that is exactly the same as the one obtained in the original =4 theory. Like in the latter case this divergence will N be cancelled by the contributions coming from the other one-loop diagrams involving V and ghost internal lines. The argument given here reinforces the results of [76], at least at the one-loop level, and supports the proposal, put forward in [77, 78], according to which the mass deformed =4 model can be considered a consistent N supersymmetry-preserving regularization for a class of =1 theories. N Notice that for the previous discussion it is not necessary to consider equal masses for all the (anti) chiral superfields. The same results can be proved giving different masses to the three superfields. This can be easily understood since in each diagram considered in this section only one chiral/anti-chiral pair is involved, because the propagators are diagonal in ‘flavour’ space and the vertices containing vector and I (anti) chiral superfields couple ΦI† and Φ with the same index I. Basically this means that the above discussed cancellations apply separately to the contributions of each (anti) chiral superfield. From the viewpoint of the dimensional analysis of [76] having different masses mI is irrelevant. As a consequence one can in particular give mass to only two of the (anti) chiral multiplets. This suggests the possibility of generalizing the approach of [77, 78] to the case of =2 super Yang–Mills theories. A discussion of the effect of a =2 mass N N term in =4 supersymmetric Yang–Mills theory can be found in [81]. N 3.2.3 Discussion

The infrared divergences found in the calculation of the propagators of the chiral and the vector superﬁelds are due to the fact that the vector superﬁeld is dimensionless, so that it contains in particular, as its lowest component, the scalar C that is itself 94 Chapter 3. Perturbative analysis of =4 SYM theory N

dimensionless and hence has a propagator which behaves, in momentum space, like 1 (CC)(k) . h i ∼ k4 The contribution of the scalar C to the VV propagator leads to an infrared diver- h i gence whenever a diagram contains a loop involving a VV line. In the Fermi–Feynman gauge the problem is not present because the choice α = 1 gives a kinetic matrix of the form of equation (3.9) in the component expansion. The corresponding inverse 1 matrix M − in (3.10) shows that no CC free propagator is present. On the con- h i 1 trary any choice α = 1 produces such a propagator, i.e. gives a matrix M − with 6 1 a non-vanishing (M − )11 entry, leading to the previously discussed problems. An explicit calculation in components with α = 1 and without fixing the Wess–Zumino 6 gauge should show problems with infrared divergences analogous to those encountered here. These problems with infrared divergences are not peculiar of the =4 super N Yang–Mills theory, but appear in any supersymmetric gauge theory. Analogous infrared divergences in Green functions have been observed in [80, 82]. The conclusion proposed in these papers is that there exist no way to remove the infrared divergences, so that the choice α=1 is somehow necessary, at least for the computation of gauge-dependent quantities. There are two general theorems concerning infrared divergences in quantum field theory. The first is the Kinoshita–Lee–Nauenberg theorem [83], which deals with infrared divergences in cross sections. It states that no infrared problem is present in physical cross sections of a renormalizable quantum field theory, if the appropriate sums over degenerate initial and final states, associated with soft and collinear particles, are considered. The second theorem was proved by Kinoshita, Poggio and Quinn [84] and concerns Green functions. The statement of the theorem is that the proper (one-particle irreducible) Euclidean Green functions with non-exceptional external momenta are free of infrared divergences in a renormalizable quantum field theory. A

set of momenta pi, i =1, 2,...,n is said to be exceptional if any of the partial sums

p , with I a subset of 1, 2,...,n , i { } i I X∈ vanishes. The reason for the absence of divergences is that the external momenta, if non-exceptional, play the rôle of an infrared regulator in the Green functions. The proof of the theorem is based on dimensional analysis which does not work in 1 the presence of a k4 propagator. In particular it fails in the case of supersymmetric 3.2 Propagators in the =1 formulation 95 N gauge theories in general gauges, because the adimensionality of the V superfield 1 implies that factors of k4 can enter the loop integrals. The apparent violation of the Kinoshita–Poggio–Quinn theorem can be understood from the viewpoint of the component formulation: supersymmetric gauge theories, being non-polynomial and thus containing an infinite number of interaction terms, are not formally renormalizable. The choice of the Wess–Zumino gauge makes the theory polynomial. In fact in this case no infrared divergence is found in Green functions and the general theorems are satisfied. However in the case of the =4 theory the choice of the WZ gauge results N in a change of the ultraviolet properties of the model, at least for what concerns gauge dependent quantities, e.g. the propagators. In theories with less supersymmetry, in which ultraviolet divergences are present the problem might seem less severe, since the subtraction of the ultraviolet infinities could cure also the infrared singularity. The infrared problems discussed here are however gauge artifacts and cannot affect gauge invariant quantities. The mechanism leading to the cancellation of infrared divergences in gauge independent Green functions has not been verified in detail yet, but can be understood starting from the super Feynman rules. When the propagator VV is inserted into a Feynman graph it is connected to vertices which carry h i covariant derivatives. These derivatives come directly from the form of the action for vertices involving only V fields and from the definition of the functional derivatives for vertices involving (anti) chiral superfields. The situation in diagrams for gauge invariant quantities is such that at least one covariant derivative can always be brought to act on the V propagator by integrations by parts. In this way an additional factor of the momentum k is generated in the numerator. More precisely for the infrared 1 2 2 2 2 problematic term (D D + D D )δ (θ θ′) one gets − k4 4 − µ 1 2 2 2 2 kµσαα˙ α˙ 2 D (D D + D D )δ (θ θ′) = 4i D D α −k4 4 − − k4 and

µαα˙ α˙ 1 2 2 2 2 kµσ 2 D (D D + D D )δ (θ θ′) =4i D D . −k4 4 − k4 α A rigorous and general check of this mechanism in gauge invariant Green functions has not been achieved yet. 96 Chapter 3. Perturbative analysis of =4 SYM theory N

3.3 Three- and four-point functions of =1 super- N ﬁelds

The computation of Green functions with three and four external legs will now be considered. From now on the Fermi–Feynman gauge will be fixed. The three-point functions should in principle suffer from infrared problems of the kind encountered in the preceding section. This claim will not be studied here since the calculation with α = 1 is awkward, even at the one-loop level, because of the huge number of 6 contributions. Four-point functions on the contrary should be infrared finite, because they are directly related to physical scattering amplitudes. Notice, however, that infrared divergences in the Green function with four external V lines were found in [82]. In that paper beyond the infrared singularity, it was shown that the adimensionality of the superfield V implies that it requires a non-linear renormalization, i.e. the

renormalized ﬁeld VR is a non-linear function of the bare ﬁeld V

VR = f(V ) .

An example of three-point function will be brieﬂy discussed here and then the more interesting case of a four-point function will be studied in greater detail.

3.3.1 Three-point functions

The simplest three point function to consider corresponds to the correction to the I J K IJK vertex εIJKtr Φ Φ Φ (or ε tr ΦI† ΦJ† ΦK† ), which is determined by one single diagram at the one loop level. However as a less trivial example of computation of three-point functions the one-loop correction to the vertex

J Φb (z)

a c I V (z) Φ †V Φ (z) c −→ I b

a ΦI†(z)

will be considered here. The correction at the one-loop level comes from the following diagrams (the notation for the propagators is the same as in the previous section) 3.3 Three- and four-point functions 97

˜ J Φb (q)

V˜ ( p q) A˜(p; q) c − − −→

˜ a ΦI†(p)

˜ J Φb (q)

V˜ ( p q) B˜(p; q) c − − −→

˜ a ΦI†(p)

˜ J Φb (q)

V˜ ( p q) C˜(p; q) c − − −→

˜ a ΦI†(p)

˜ J Φb (q)

V˜ ( p q) D˜(p; q) c − − −→

˜ a ΦI†(p)

˜ J Φb (q)

V˜ ( p q) E˜(p; q) c − − −→

˜ a ΦI†(p) 98 Chapter 3. Perturbative analysis of =4 SYM theory N

˜ J Φb (q)

V˜ ( p q) F˜(p; q) c − − −→

˜ a ΦI†(p)

In the presence of a mass term for the (anti) chiral superﬁelds the expressions given above are modiﬁed by the presence of the mass in the free propagators and furthermore there are two additional sets of contributions corresponding to the diagrams

˜ J Φb (q)

V˜ ( p q) G˜(p; q) c − − −→

˜ a ΦI†(p) ˜ J Φb (q)

V˜ ( p q) H˜ (p; q) c − − −→

˜ a ΦI†(p)

The calculation of these diagrams is completely analogous to those that were presented in the previous section. All of these proper contributions can be derived using the Feynman rules. The last two diagrams are finite and vanish in the =4 theory, N i.e. at m=0, as they are proportional to m2; they will not be considered here. The other contributions will be briefly studied in the limit m 0. → The diagram D˜(p, q) is zero as a consequence of the contraction among the colour indices, so that there are five contributions to be calculated. The dimensional analysis of section 1.7 gives a vanishing superficial degree of divergence, corresponding to a logarithmic divergence, for the Green function under consideration. Each diagram actually contains a logarithmically divergent term plus finite terms. A straightforward but rather lengthy computation, based on elementary D-algebra, allows to prove that 3.3 Three- and four-point functions 99

the divergent part of all the diagrams is of the form

4 I abc d k 4 1 J I = c δ f d θ Φ† (p,θ, θ)V ( p q,θ, θ)Φ (q,θ, θ) , log J (2π)4 k2(p k)2 aI b − − c Z − n o where c is a constant. Moreover one finds, from each diagram a finite contribution of the form d4k 1 I = δI f abc d4θ finite J (2π)4 k2(q + k)2(p k)2 Z − α 2 J Φ† (p,θ, θ) c D D D V ( p q,θ, θ) Φ (q,θ, θ) + aI 1 α b − − c µ n h α˙ i +c σ (q + k) DαD V ( p q,θ, θ) ΦJ (q,θ, θ) , 2 αα˙ µ b − − c h i o where c1 and c2 are numerical constants. The sum of all the logarithmically divergent terms contained in the diagrams A˜, B˜, C˜, E˜ and F˜ vanishes. The residual finite part can be shown to be zero as well. In conclusion the one-loop correction to the three- point function Φ†V Φ exactly vanishes in the Fermi–Feynman gauge. The same h i result can be shown to hold for the other three-point functions. The superfield formalism does not lead to significant simplifications in the calculation of three-point functions with respect to the same computation in components (in the Wess–Zumino gauge!); actually for the Green function considered here the number of diagrams to be evaluated is approximately the same in the component formulation. However the power of superspace techniques becomes clear in the computation of four-point functions that will be considered in next subsection.

3.3.2 Four-point functions

The computation of four-point functions in components is awkward even at the one- loop level and in the Wess–Zumino gauge. In this case the choice of the WZ gauge should not lead to divergences because the complete four-point function must finally give the physical scattering amplitude. In the =1 superfield formulation the calcu- N lation of four-point functions, though rather lengthy, is much more simple. In this section the computation of the one-particle irreducible one-loop correction to the Green function Φ†ΦΦ†Φ in the Fermi–Feynman gauge will be presented. h i There are several diagrams to be considered: each of them is free of ultraviolet divergences, as immediately follows from dimensional analysis. Moreover in the Fermi– Feynman gauge the single corrections are infrared safe. In conclusion one finds a finite and non-vanishing result. 100 Chapter 3. Perturbative analysis of =4 SYM theory N

The ﬁrst subset of diagrams corresponds to

˜ J ˜ c Φb ΦK†

A˜(p; q; r) −→

˜ a ˜ L ΦI† Φd

together with those obtained from this one by crossing. There are a total of four diagrams of this kind. Application of the Feynman rules and steps completely analogous to those entering the evaluation of the propagators allow to ﬁnd the following results. 1 2 A˜ (p,q,r)= g4 κN 2(N 2 1)(δbδd + δ δbd)δI δKI(A)acJL(p,q,r) 1 4 − a c ac J L 1 bdIK where d4k 1 I(A)acJL(p,q,r) = d4θ 1 bdIK (2π)4 k2(p k)2(q + k)2(q + k r)2 · Z − − 2 2 a c L J a (D D )Φ †(p,θ, θ) Φ †(p + q r, θ, θ)Φ (r, θ, θ)Φ (q,θ, θ) + Φ †(p,θ, θ) · I K − d b I · nh 2 2 c i L J 2 a (D D )Φ †(p + q r, θ, θ) Φ (r, θ, θ)Φ (q,θ, θ)+2 (D D )Φ †(p,θ, θ) · K − d b α˙ I · h α˙ c L i J h a i D Φ †(p + q r, θ, θ) Φ (r, θ, θ)Φ (q,θ, θ)+2 D Φ †(p,θ, θ) · − d b α˙ I · h 2 α˙ c i L J h α ia (D D )Φ †(p + q r, θ, θ) Φ (r, θ, θ)Φ (q,θ, θ)+4 (D D )Φ †(p,θ, θ) · − d b α˙ I · h α˙ c i L J h i D D Φ †(p + q r, θ, θ) Φ (r, θ, θ)Φ (q,θ, θ) · α − d b ≡ h d4k i 1 o d4θ KacJL(p,q,r; θ; θ) (3.33) ≡ (2π)4 k2(p k)2(q + k)2(q + k r)2 bdIK Z − − and the constant κ is a group theory factor deﬁned by f f bfgf f dhe = κN 2(N 2 1)(δbδd + δdδb) ; aef cgh − a c a c 1 2 A˜ (p,q,r)= g4 κN 2(N 2 1)(δdδb + δ δbd)δI δKI(A)acJL(p,q,r) , 2 4 − a c ac L J 2 bdIK where d4k 1 I(A)acJL(p,q,r) = d4θ 2 bdIK (2π)4 k2(p k)2(p k r)2(q + k)2(p + q k r)2 · Z − − − − − KacJL(p,q,r; θ; θ); (3.34) · bdIK 3.3 Three- and four-point functions 101

1 2 A˜ (p,q,r)= g4 κN 2(N 2 1)(δbδd + δdδb)δI δKI(A)acJL(p,q,r) , 3 4 − a c a c J L 3 bdIK where d4k 1 I(A)acJL(p,q,r) = d4θ 3 bdIK (2π)4 k2(p k)2(q + k)2(q + k r)2 · Z − − 2 a 2 J c L D Φ †(p,θ, θ) D Φ (q,θ, θ) Φ †(p + q r, θ, θ)Φ (r, θ, θ)+ · I b K − d nh µ i α˙ a α J c +8iσ (p k r) D Φ †(p,θ,θ) D Φ (q,θ, θ) Φ †(p + q r, θ, θ) αα˙ − − µ I b K − · L 2 a J c Φ (r, θ, θ) 16(p hk r) Φ †(p,θ,i θ)Φ (q,θ, θ)Φ†(p + q r, θ, θ) · d − − − I b K − · ΦL(r, θ, θ) ; (3.35) · d

1 2 A˜ (p,q,r)= g4 κN 2(N 2 1)(δbδd + δdδb)δI δKI(A)acJL(p,q,r) , 4 4 − a c a c J L 4 bdIK where d4k 1 I(A)acJL(p,q,r) = d4θ 4 bdIK (2π)4 k2(p k)2(p k r)2(q + k)2 · Z − − − 2 a J c 2 L D Φ †(p,θ, θ) Φ (q,θ, θ)Φ †(p + q r, θ, θ) D Φ (r, θ, θ) + · I b K − d nh µ i α˙ a J c +8iσ (q + k) D Φ †(p,θ, θ) Φ (q,θ, θ)Φ †(p+ q r, θ, θ) αα˙ µ I b K − · α L 2 a J c D Φ (r, θ, θ) h 16(q + k) Φi†(p,θ, θ)Φ (q,θ, θ)Φ †(p + q r, θ, θ) · d − I b K − · Φ L(r, θ, θ) . (3.36) · d The second kind of contribution corresponds to the diagram

˜ J ˜ c Φb ΦK†

B˜(p; q; r) −→

˜ a ˜ L ΦI† Φd

In this case the diagrams obtained by crossing are identical to the one depicted, so that they are accounted for by giving the correct weight to B˜(p,q,r). For this contribution one ﬁnds g4 B˜(p,q,r)= κN 2(N 2 1)(δbδd + δdδb)(δI δK + δI δK)I(B)acJL(p,q,r) , 6 − a c a c J L L J bdIK 102 Chapter 3. Perturbative analysis of =4 SYM theory N

where d4k 1 I(B)acJL(p,q,r) = d4θ bdIK (2π)4 k2(p k)2(p + q r)2(q + k)2 · Z − − 2 a J c 2 L D Φ †(p,θ, θ) Φ (q,θ, θ)Φ †(p + q r, θ, θ) D Φ (r, θ, θ) + · I b K − d nh µ i α˙ a J c +8iσ (p k) D Φ †(p,θ, θ) Φ (q,θ, θ)Φ †(p+ q r, θ, θ) αα˙ − µ I b K − · α L 2 a J c D Φ (r, θ, θ) h 16(p k) Φi†(p,θ, θ)Φ (q,θ, θ)Φ †(p + q r, θ, θ) · d − − I b K − · ΦL(r, θ, θ) . (3.37) · d

The next one-loop correction is

˜ J ˜ c Φb ΦK†

C˜(p; q; r) −→

˜ a ˜ L ΦI† Φd

This contribution is trivially zero because it contains the product

δ (θ θ )δ (θ θ ) 0 . 4 1 − 2 4 1 − 2 ≡ This would not be true with a choice of gauge diﬀerent from the Fermi–Feynman one, i.e. with α = 1, because in that case there would be projectors acting on the δ’s. 6 The vanishing of C˜(p,q,r), that is completely trivial in the superﬁeld formulation, corresponds, in the component formulation, to a complicated cancellation among various terms coming from graphs with the same topology. Another subset of diagrams contains

˜ c ˜ J ΦK† Φb

D˜(p; q; r) −→

˜ a ˜ L ΦI† Φd 3.3 Three- and four-point functions 103

plus the crossed versions of this one. There are three inequivalent crossed diagrams. The result of the calculation is the following.

1 2 D˜ (p,q,r)= g4 κN 2(N 2 1)(δ δbd + δdδb)(δI δK δI δK)I(D)acJL(p,q,r) , 1 4 − ac a c J L − L J 1 bdIK where d4k 1 I(D)acJL(p,q,r) = d4θ 1 bdIK (2π)4 k2(p k)2(k + r p q)2(p k r)2 · Z − − − − − 2 a 2 J c L D Φ †(p,θ, θ) D Φ (q,θ, θ) Φ †(p + q r, θ, θ)Φ (r, θ, θ)+ · I b K − d nh µ i α˙ a α J c +8iσ (k + r p) D Φ †(p,θ,θ) D Φ (q,θ, θ) Φ †(p + q r, θ, θ) αα˙ − µ I b K − · L 2 a J c Φ (r, θ, θ) 16(k +h r p) Φ †(p,θ,i θ)Φ (q,θ, θ)Φ†(p + q r, θ, θ) · d − − I b K − · ΦL(r, θ, θ) ; (3.38) · d

1 2 D˜ (p,q,r)= g4 κN 2(N 2 1)(δ δbd + δbδd)(δI δK δI δK)I(D)acJL(p,q,r) , 2 4 − ac a c L J − J L 2 bdIK where d4k 1 I(D)acJL(p,q,r) = d4θ 2 bdIK (2π)4 k2(p k)2(k + r p q)2(p + q k)2 · Z − − − − 2 a J c 2 L D Φ †(p,θ, θ) Φ (q,θ, θ)Φ †(p + q r, θ, θ) D Φ (r, θ, θ) + · I b K − d nh µ i α˙ a J c +8iσ (k p q) D Φ †(p,θ, θ) Φ (q,θ, θ)Φ †(p + q r, θ, θ) αα˙ − − µ I b K − · α L 2 a J c D Φ (r, θ, θ) 16(h k p q) Φi†(p,θ, θ)Φ (q,θ, θ)Φ †(p + q r, θ, θ) · d − − − I b K − · Φ L(r, θ, θ) ; (3.39) · d

1 2 D˜ (p,q,r)= g4 κN 2(N 2 1)(δdδb + δ δbd)(δI δK δI δK)I(D)acJL(p,q,r) , 3 4 − a c ac J L − L J 3 bdIK where d4k 1 I(D)acJL(p,q,r) = d4θ 3 bdIK (2π)4 k2(p k)2(k + r p q)2(p + q k)2 · Z − − − − a J 2 c 2 L Φ †(p,θ, θ)Φ (q,θ, θ) D Φ †(p + q r, θ, θ) D Φ (r, θ, θ) + · I b K − d n µ a h J α˙ ic 8iσ (p k r) Φ †(p,θ, θ)Φ (q,θ, θ) D Φ †(p + q r, θ, θ) − αα˙ − − µ I b K − · α L 2 a J c D Φ (r, θ, θ) 16(p k r) Φ †(p,θ,hθ)Φ (q,θ, θ)Φ †(p + qi r, θ, θ) · d − − − I b K − · ΦL(r, θ, θ) ; (3.40) · d

104 Chapter 3. Perturbative analysis of =4 SYM theory N

1 2 D˜ (p,q,r)= g4 κN 2(N 2 1)(δbδd + δ δbd)(δI δK δI δK)I(D)acJL(p,q,r) , 4 4 − a c ac L J − J L 4 bdIK where d4k 1 I(D)acJL(p,q,r) = d4θ 4 bdIK (2π)4 k2(p k)2(k + r p q)2(p + q k)2 · Z − − − − a 2 J 2 c L Φ †(p,θ, θ) D Φ (q,θ, θ) D Φ †(p + q r, θ, θ) Φ (r, θ, θ)+ · I b K − d n µ a h α J α˙ ic 8iσ (p + q k) Φ †(p,θ,θ) D Φ (q,θ, θ) D Φ †(p + q r, θ, θ) − αα˙ − µ I b K − · L 2 a J c Φ (r, θ, θ) 16(p + q k) Φ †(p,θ, θ)Φ (q,θ, hθ)Φ †(p + q r, θ, θ) i · d − − I b K − · ΦL(r, θ, θ) . (3.41) · d The last one-loop family of diagrams is formed by the one below plus those obtained by crossing.

˜ J ˜ c Φb ΦK†

E˜(p; q; r) −→

˜ a ˜ L ΦI† Φd

The resulting contributions to the Green function are 1 2 g4 E˜ (p,q,r)= κN 2(N 2 1)δbδd + δ δbd)δI δK I(E)acJL(p,q,r) , 1 4 2 − a c ac J L 1 bdIK where d4k 1 I(E)acJL(p,q,r) = d4θ 1 bdIK (2π)4 k2(p k)2(p + q k)2 · Z − − a J c L Φ †(p,θ, θ)Φ (q,θ, θ)Φ †(p + q r, θ, θ)Φ (r, θ, θ) ; (3.42) · I b K − d n o 1 2 g4 E˜ (p,q,r)= κN 2(N 2 1)(δdδb + δ δbd)δI δK I(E)acJL(p,q,r) , 2 4 2 − a c ac L J 2 bdIK where d4k 1 I(E)acJL(p,q,r) = d4θ 2 bdIK (2π)4 k2(p k)2(r k)2 · Z − − a J c L Φ †(p,θ, θ)Φ (q,θ, θ)Φ †(p + q r, θ, θ)Φ (r, θ, θ) ; (3.43) · I b K − d n o 3.3 Three- and four-point functions 105

1 2 g4 E˜ (p,q,r)= κN 2(N 2 1)(δbδd + δ δbd)δI δKI(E)acJL(p,q,r) , 3 4 2 − a c ac L J 3 bdIK where d4k 1 I(E)acJL(p,q,r) = d4θ 3 bdIK (2π)4 k2(p + q k)2(k r)2 · Z − − a J c L Φ †(p,θ, θ)Φ (q,θ, θ)Φ †(p + q r, θ, θ)Φ (r, θ, θ) ; (3.44) · I b K − d n o 1 2 g4 E˜ (p,q,r)= κN 2(N 2 1)(δdδb + δ δbd)δI δK I(E)acJL(p,q,r) , 4 4 2 − a c ac L J 4 bdIK where d4k 1 I(E)acJL(p,q,r) = d4θ 4 bdIK (2π)4 k2(q + k)2(k + r p)2 · Z − a J c L Φ †(p,θ, θ)Φ (q,θ, θ)Φ †(p + q r, θ, θ)Φ (r, θ, θ) . (3.45) · I b K − d The sum of alln the preceding terms results in a finite and non vaonishing total one-loop correction to the four point function. The final expression contains terms with six different tensorial structures 2 6 1 4 2 2 (i) Φ†ΦΦ†Φ = κ g N (N 1) G , h i 4 − i=1 X where 2 G(1) = δbδdδI δK I(A)acJL + I(A)acJL + I(B)acJL I(D)acJL+ a c J L 1 bdIK 3 bdIK 3 bdIK − 2 bdIK I(D)acJL + I(E)acJL + I(E)acJL − 4 bdIK 1 bdIK 3 bdIK (A)acJL (D)acJL (D)acJL (D)acJL G(2) = δ δbdδI δK I + I I + I + ac J L 1 bdIK 1 bdIK − 2 bdIK 3 bdIK (D)acJL (E)acJL (E)acJL I + I + I − 4 bdIK 1 bdIK 3 bdIK 2 G(3) = δbδdδI δK I(A)acJL + I(B)acJL + I(D)acJL + I(D)acJL a c L J 2 bdIK 3 bdIK 2 bdIK 4 bdIK 2 G(4) = δdδbδI δK I(A)acJL + I(A)acJL + I(B)acJL I(D)acJL a c L J 2 bdIK 4 bdIK 3 bdIK − 1 bdIK − I(D)acJL + I(E)acJL + I(E)acJL − 3 bdIK 2 bdIK 4 bdIK 2 G(5) = δdδbδI δK I(A)acJL + I(B)acJL + I(D)acJL + I(E)acJL a c J L 3 bdIK 3 bdIK 1 bdIK 3 bdIK G(6) = δ δbdδI δK I(A)acJL I(D)acJL + I(D)acJL I(D)acJL+ ac L J 4 bdIK − 1 bdIK 2 bdIK − 3 bdIK (D)acJL (E)acJL (E)acJL + I4 bdIK + I2 bdIK + I4 bdIK 106 Chapter 3. Perturbative analysis of =4 SYM theory N

In the presence of a mass term for the (anti) chiral superﬁelds the expressions given above are slightly modiﬁed by the presence of the mass in the free propagators and furthermore there are two additional sets of contributions corresponding to the diagrams

˜ c ˜ L ΦK† Φd

F˜(p; q; r) −→

˜ a ˜ J ΦI† Φb

˜ J ˜ L Φb Φd

G˜(p; q; r) −→

˜ a ˜ c ΦI† ΦK†

Both of these graphs give corrections proportional to m2, that can be calculated much in the same way as the previous ones.

With a diﬀerent choice of gauge the computation of four-point Green functions is more complicated. Single diagrams involving vector superﬁeld propagators contain new contributions, some of which are infrared divergent. The correction C˜ is not zero anymore, because there are projection operators acting on the δ-functions. Moreover further diagrams must be included in the calculation at the same order as a consequence of the non-vanishing of the one-loop correction to the vertices. For example one must consider diagrams such as 3.3 Three- and four-point functions 107

˜ J ˜ L Φb Φd

˜ a ˜ c ΦI† ΦK†

The techniques illustrated in this chapter for the calculation of Green functions in the =1 formalism can be applied to correlation functions of composite operators N as well. Green functions of gauge invariant composite operators such as those that form the multiplet of currents (equations (2.17) and (2.18)) play a crucial rôle in the correspondence with type IIB superstring theory on AdS space to be discussed in chapter 5. The application of =1 superspace to this problem will be sketched in N the final chapter. It can be shown that the extension of the formalism to the case of composite operators is rather straightforward, the fundamental difference being that the complete Green functions and not the proper parts must be considered. Chapter 4

Nonperturbative eﬀects in =4 super Yang–Mills theory N

In the last chapter the quantum properties of =4 supersymmetric Yang–Mills the- N ory have been analyzed at the perturbative level, this chapter is devoted to the study of some non-perturbative effects. Instanton calculus will be employed to compute various Green functions of gauge invariant composite operators in the semiclassical approximation. It is well known that in gauge field theories non-perturbative effects can modify in a dramatic way the perturbative structure of the vacuum. Instanton calculus has proved to be an extremely powerful tool in the analysis of non-perturbative phenomena in quantum field theories, particularly in the case of supersymmetric gauge theories. The rôle of instantons in Yang–Mills theories was pointed out in [85] at the classical level and then at the quantum level in a fundamental work by ‘t Hooft [86]. Instantons are non trivial finite action solutions of the classical Euclidean equations of motion of the model. The computation of Green functions in the instanton background allows to study non-perturbative properties of the quantum theory. In non-supersymmetric non Abelian gauge theories the contribution of instantons to Green functions turns out to be divergent in the semiclassical approximation [86]. However in the supersymmetric case explicit instanton calculations can be performed leading to finite and well defined results, thanks to the cancellation between fermionic and bosonic quantum fluctuations. Compensations of quantum effects due

108 109

bosons and fermions take place in the instanton background and allow to explicitly compute vacuum expectation values (vev’s) of composite operators (condensates). These cancellations are reminiscent of boson and fermion compensations in internal loops at the perturbative level. The condensates play the rôle of order parameters in non-perturbative phenomena such as for instance the spontaneous breaking of chiral symmetries or even of supersymmetry in certain models [87]. Furthermore instanton calculus together with holomorphy properties, has allowed to derive some exact results in supersymmetric theories [33] and, more recently, to obtain non trivial checks of the exact result proposed in [34]. As was discussed in previous chapters supersymmetric gauge theories typically possess a moduli space of vacua parameterized by the vev’s of scalar fields. The literature on instanton effects in asymptotic free theories includes examples in which the computations are performed with zero vev’s for all the scalars [87] as well as examples in which non vanishing vev’s for the scalar fields parameterizing the moduli space are considered [88]. The =4 super Yang–Mills theory has a (exactly) vanish- N ing β-function and a moduli space of vacua, =R6k , where k is the rank of the M × Sk gauge group and the group of permutations of k elements, parameterized by the Sk vev’s of six real scalar fields, ϕi . The superconformal phase, corresponding to the h i origin of , i.e. ϕi =0, i, cannot be obtained naively as the limit of the theory M h i ∀ in the Coulomb phase. In this chapter the approach of [87] will be applied to the computation of various condensates corresponding to operators in the supermultiplet of currents (2.17), (2.18). Four-, eight- and sixteen-point functions of bilinear operators belonging to the current supermultiplet will be computed in the semiclassical approximation in the case of a SU(2) gauge group. In particular the exact complete spatial dependence will be determined for a four-point function at lowest order. Due to the vanishing of the β-function and of the associated chiral anomalies, the correlation functions that will be studied, unlike the cases encountered in theories with less supersymmetry, receive a non-vanishing perturbative contribution as well. These Green functions will reconsidered in the final chapter in the context of the correspondence with type IIB superstring theory compactified on AdS S5. It will 5 × be shown that the difficulties encountered in perturbative calculations in chapter 3 do not affect the computations discussed here. The chapter is organized as follows. The general features of instanton calculus in supersymmetric gauge theories are briefly discussed in sections 1, 2 and 3. Sections 4.4, 4.5 and 4.6 report on original results in the case of =4 supersymmetric Yang– N 110 Chapter 4. Non-perturbative effects in =4 SYM theory N

Mills theory, following [89]. The ﬁnal section describes the generalizations of the results of [89] to the case of a SU(N) gauge group and to the K-instanton sector in the large N limit, that were given respectively in [90] and [91, 92].

4.1 Instanton calculus in non Abelian gauge theories

In this section the semiclassical quantization in the background of an instanton configuration will be reviewed for the case of a pure Yang–Mills theory. The extension to fermions will be discussed in the next section. In quantum field theory one is interested in computing vacuum expectation values of operators (φ), that in general are polynomials in the elementary fields φ. In O Euclidean space one must calculate

S[φ] (x ,...,x ) = [ φ]e− ℏ (φ(x ),...,φ(x )) , (4.1) hO 1 n i D O 1 n Z where the dependence on ℏ has been explicitly indicated in order to discuss the semiclassical approximation. The classical vacuum of the theory corresponds to a constant expectation value φ for the elementary fields and a perturbative series is obtained expanding the h i interaction part of the action S[φ] for small fluctuations, δφ, around the classical vacuum. In instanton calculations one considers a situation in which there exists a non-trivial solution of the classical equations of motion with finite Euclidean action. Denoting by φ such a field configuration one can then study quantum fluctuation around φ = φ in the limit ℏ 0. Putting →

φ(x)= φ(x)+ δφ(x)= φ(x)+ ℏ1/2η(x)

and expanding S[φ] in powers of ℏ in the form

ℏ δ2S S[φ]= S[φ]+ d4xd4y η(x)η(y)+ O(ℏ3/2) , 2 δφ(x)δφ(y) Z φ=φ

the vacuum expectation value of an operator [φ] can be evaluated in the semiclas- O 4.1 Instantons in Yang–Mills theories 111

sical limit by a saddle point approximation 1

S[φ+η] (x ,...,x ) = [ η]e− ℏ [(φ + η)(x )] = hO 1 n i D O i Z 1 2 − 2 S δ S = (φ) e− ℏ det (1 + O(ℏ)) . (4.2) O δφ(x)δφ(y) " φ=φ#

Notice that in principle one could consider a classical solution φ with inﬁnite Eu- clidean action, but this would give a vanishing contribution in the semiclassical ap- S[φ]/ℏ proximation because of the factor e− in (4.2).

4.1.1 Yang–Mills instantons

The classiﬁcation of non-trivial solutions of the classical Euclidean equations of motion for the Yang–Mills system has been given in [85], where an explicit solution has also been constructed. Another (multi-center) solution has been proposed by ’t Hooft, who also studied the semiclassical quantization [86]. The case of a SU(2) gauge group will be discussed ﬁrst and then the extension to SU(N) will be sketched. The Euclidean Yang–Mills action is

1 4 µν SYM[A]= d x tr(Fµν F ) , (4.3) −4drg2 Z where the gauge ﬁeld Aµ and the ﬁeld strength Fµν are respectively

a Aµ = AµTa F µν = ∂µAν ∂ν Aµ + [Aµ, Aν]= ∂ Aa ∂ Aa + if a Ab Ac T − µ µ − ν µ bc µ ν a and T a are generators of the gauge group in the representation r, satisfying [T a, T b]= ab c a b ab if cT and tr T T = δ dr, dr being the Dynkin index of the representation. Gauge transformations take the form

Ω A (x) A (x)=Ω(x) [A (x)+ i∂ ] Ω†(x) , µ −→ µ µ µ

a with Ω(x)= eiαa(x)T . The study of finite Euclidean action configurations is performed much in the same way as for the monopoles of the Georgi–Glashow model in appendix B. The requirement of finite action implies that Aµ must approach asymptotically, i.e. for 1This relation holds for bosonic fields φ, see the next section for the case of fermions. 112 Chapter 4. Non-perturbative effects in =4 SYM theory N

x , a pure gauge configuration. This leads to a topological classification of the | |→∞ field configurations in homotopy classes. The quantity which characterizes the finite energy solutions is the Pontryagin index

1 K[A]= d4xF a F˜µν , (4.4) 32π2 µν a Z ˜ 1 ρσ where F = 2 εµνρσF . The Yang–Mills action satisﬁes a “Bogomol’nyi bound”

8π2 1 8π2 S [A]= K[A]+ d4x F a F˜a F µν F˜µν K[A] . (4.5) YM ± g2 8π2 µν ∓ µν a ∓ a ≥ g2 | | Z Solutions of the classical equations of motion correspond to local minima of the action functional, so that in each homotopy class they saturate the bound in (4.5). As a consequence they are determined by the condition

F a = F˜a . µν ± µν

In the K=0 sector the minimum is reached on the trivial vacuum, Aµ=0, while the conﬁgurations minimizing S[A] for K = 0 are (anti) instantons. Self-dual and anti | | 6 self-dual conﬁgurations are referred to as respectively instantons and anti-instantons. In the K=1 sector and for a gauge group SU(2) the explicit solution was found in [85, 86] through the ansatz

σ A = i µν ∂ν log Φ , (4.6) µ 2 where Φ is real scalar ﬁeld. Substituting into the equations of motion one obtains the equation for Φ

2Φ =0 . Φ A solution can be written in the form ρ2 Φ=1+ . (4.7) (x x )2 − 0

Φ depends on 5 free parameters, ρ associated with the “size” of the instanton and x0 with its position. The corresponding instanton solution is yν A = 2iρ2ηa , (4.8) µ − µν y2(y2 + ρ2) 4.1 Instantons in Yang–Mills theories 113

where y = x x and ηa are the anti self-dual ’t Hooft symbols − 0 µν εaµν for µ, ν =1, 2, 3 ηa = ηa = µν − νµ δaµ for ν =4 . − The general solution depends on 3 additional parameters corresponding to the freedom of performing global gauge rotations. Actually the Yang–Mills system is classi- cally invariant under the four-dimensional conformal group, which partially overlaps with global gauge transformations. In [93] it has been proved that the most general one-instanton solution depends on 8 parameters, referred to as collective coordinates or moduli, that in the following will be denoted by βi. The moduli space, i.e. the space parameterized by the coordinates β , will be indicated by . The ﬁeld (4.8) i M has a singularity at y=0, which can be removed by a correspondingly singular gauge transformation, so that one can construct a solution with no singularities at any x

ν a a iθaT a y iθaT A = 2ie η e− , (4.9) µ − µν (y2 + ρ2)

a where ηµν are the self-dual ’t Hooft symbols

εaµν for µ, ν =1, 2, 3 ηa = ηa = µν − νµ δaµ for ν =4 , which satisfy

a σµν = ηµνσa , with σa, a =1, 2, 3 the standard Pauli matrices. The associated ﬁeld strength is

2 a a iθaT ρ iθaT F = F [A]= 2ie σ e− . (4.10) µν µν − µν (y2 + ρ2)2 The ansatz of (4.7) can be generalized to a n-instanton solution considering the scalar ﬁeld n ρ2 Φ(x)=1+ i . (i) 2 i=1 (x x0 ) X − (n) (i) Substituting into (4.6) yields a solution, Aµ , with singularities at xµ = x0 µ, which can be made regular by a suitable singular gauge transformation. For such a conﬁg- uration one can prove that

(n) (1) S[A ]= nS[A ] , 114 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

(n) so that Aµ (x) is a solution in the homotopy class K[A]=n. The number of collective coordinates for the n-instanton solution is 8n 3 (with n > 1). There are 8 moduli − for each single instanton, but one must subtract 3 global rotations that are already accounted for by the relative rotations. This result can be proved to hold in general, even if the n instantons are not widely separated, [94].

4.1.2 Semiclassical quantization

To perform a semiclassical quantization around an instanton conﬁguration according

to (4.2) one needs the expansion of S[A] around Aµ = Aµ, obtained putting Aµ =

Aµ + Qµ, where Qµ denotes the quantum ﬂuctuation. A straightforward calculation gives the gauge kinetic operator in the instanton background δ2S[A] M ab = = D2(A)δ + D (A)D (A) 2F δab , µν δAa δAb − µν µ ν − µν µ ν A

where Dµ(A)= ∂µ + i[Aµ, . ] is the covariant derivative. In the instanton background the Faddeev–Popov quantization must be suitably

modiﬁed, because the kinetic operator Mµν possesses 8 new zero modes (in the case of a SU(2) gauge group under consideration) beyond those associated with gauge symmetry. These new zero-modes correspond to the symmetries of the action S[A] that are broken by the instanton conﬁguration and are related to the 8 collective

coordinates βi in the general solution (4.9). In fact since Aµ is a solution for any βi

δ2S[A] ∂A d4y ν (y, β)=0 , δAµδAν ∂βi Z Aβ

so that (4.2) is formally divergent. This divergence is completely analogous to those introduced in the partition function of gauge theories by the functional integration along the gauge orbits, that is dealt with by the Faddeev–Popov procedure. The correct way of performing the quantization in the instanton background was discussed in [95]. The general instanton conﬁguration will be parameterized as

Ω Aµ (x, β) = Ω†AµΩ+ iΩ†∂µΩ ,

where Ω denotes an arbitrary gauge transformation and β the set collective coordinates, 8 in the case of SU(2) and K[A]=1 and in general 4NK[A] in the K-instanton sector for a SU(N) gauge group. The basic idea is then to constrain `ala Faddeev– Ω Popov the quantum ﬂuctuations Qµ to be orthogonal to Aµ as Ω varies in G and βi 4.1 Instantons in Yang–Mills theories 115

in the moduli space . In the partition function of the system 2 M

S[A] Z = [ A]e− D Z one inserts

Ω Ω Ω δA Ω ∂A 1 = ∆ [ ha(x)] dβ δ (A A , µ ) δ (A A , µ ) , FP D i µ − µ δha µ − µ ∂β G a,x i ! i ! Z Y ZM Y (4.11)

a where Ω(x) is parameterized as Ω(x) = eih (x)Ta and the scalar product in the arguments of the δ-functions is 1 (f ,gµ)= d4x tr(f (x)gµ(x)) . (4.12) µ 2 µ Z

Equation (4.11) deﬁnes the Faddeev–Popov determinant ∆FP. Some algebraic steps allow write Z in the form

S[A] Z = [ A ] dβ e− ∆ (A, β) D µ i FP · i Z Z Y ∂A δ tr T aD (A) Aµ A (x) δ (A A , µ ) . · µ − µ − µ ∂β i In this relation an inﬁnite constant, coming from the integration over the group algebra, has been dropped, since in the following the above construction will be employed in the computation of normalized vev’s of operators. The Faddeev–Popov determinant in the above expression can be computed much in the same way as in the case of the ordinary of gauge theories. To compute expectation values in the semiclassical approximation one only needs

∆FP(A, β) on the classical solution A. It can be shown that in this case ∆FP can be written as

2 ∆FP(A, β) = det Dab(A)δ(x y) det (dij(A, β)) , (4.13) − i,j A=A x,y a,b

where

(i) µ(j) dij(A, β)=(âµ (β), â (β)) 2From now on ℏ will be set equal 1. 116 Chapter 4. Non-perturbative effects in =4 SYM theory N

The vectorsa ˆ in the above equation are given by a (i) a ∂Aµ (i) a aˆµ (x, β)= + ξµ (x) ∂βi (i) a and ξµ ) are determined by the transversality condition

ab (i) Dµ (A)ˆaµ b(x, β)=0 . In conclusion introducing ghost ﬁelds, c and c to rewrite the determinants as Gaussian integrals the expectation value of a gauge invariant operator [A] in the O semiclassical approximation takes the form

2 8π K[A] (x ,...,x ) = e− g2 hO 1 n iA · 1 µ g.f. ν 2 aˆ Q Mµν Q cD (A)c [ Q c c] dβ k k e− 2 − [A] D µD D i √ O i 2π R R Z Y = 1 µ (0)g.f. ν · Q Mµν Q c2c [ Qµ c c]e− 2 − D D D R R Z 1 2 g.f. − 2 2 8π K[A] aˆ det′ Mµν det D (A) − g2 = e dβi k k [A] 1 , (4.14) √ O 2 i 2π (0)g .f. − 2 Z Y det′ Mµν det g.f. (0)g.f. where the ‘gauge-ﬁxed’ kinetic operators Mµν and Mµν are M g.f. = D2(A)δ 2F [A] , M (0)g.f. = 2δ . µν − µν − µν µν − µν 4.1.3 Bosonic zero-modes

To explicitly compute expectation values through equation (4.14) one needs to know (i) the zero-mode vectorsa ˆµ , whose norms enter the integration measure. Since the (i) kinetic operator is modiﬁed by the gauge-ﬁxing term the vectorsa ˆµ are given by

(i) ∂Aµ (i) aˆµ (x)= (x, β)+ Dµ(A)Λ (x) , ∂βi for a suitable function Λ(i)(x), such that

aˆ(i)(x) Ker M g.f. . µ ∈ µν To compute the zero-mode vectors it is convenient to refer to the singular instanton solution. For the case of a SU(2) gauge group, after rescaling A 1 A , it reads µ → g µ 2 a a i iθaT ρ (x x0)µ iθaT A (x, β)= e σ − e− , (4.15) µ −g µν (x x )2[(x x )2 + ρ2] − 0 − 0 4.1 Instantons in Yang–Mills theories 117

µ where the 8 moduli are βi =(x0 , ρ, θa).

Zero-modes associated with translations.

In this case β xµ and there are four zero-modes. The transverse vectors are i → 0 ∂ aˆµ(ν) = ν Aµ + Dµ(A)Aν = Fµν [A] . ∂x0 The computation of the norm is straightforward using (4.15), the result is

2√2 π aˆ = . k µ(ν)k g

Zero-modes associated with dilatations.

(dil) The corresponding collective coordinate is the size of the instanton ρ.a ˆµ is simply given by

∂A 2i ρyν aˆ(dil) = µ = σ , µ ∂ρ − g µν (y2 + ρ2)2 so that for the norm one obtains 4π aˆ(dil) = . k µ k g

Zero-modes associated with gauge rotations.

The relevant collective coordinates are the angles θa in (4.15) and one can put

∂A i aˆ(a) = µ + D (A)Λ(a) = D (A) T a +Λ(a) D (A)φ(a) , µ ∂θ µ µ −g ≡ µ a where the scalar field φ(a), defined by the last equality in the above equation, must satisfy i D2(A)φ(a) =0 , lim φ(a) = T (a) . x −g | |→∞ The solution is i r2 φ(a) = T (a) with r = x . −g r2 + ρ2 | | 118 Chapter 4. Non-perturbative effects in =4 SYM theory N

The norms are then 2πρ aˆ(a) = . k µ k g The above eight vectors can be shown to be orthogonal with respect to the scalar product (4.12).

Having determined the vectorsa ˆµ one can thus replace the integration over the zero-modes in (4.14) with an integration over the instanton collective coordinates and write the integration measure as 4 aˆ 1 2√2π 4π 2πρ 3 dµ = dβ k k = d4x dρd3θ = B i √ (2π)4 g g g 0 i 2π ! Y 27π4 = ρ3d4x dρd3θ . (4.16) g8 0 The generalization to larger gauge groups, in particular to SU(N), is rather straightforward [96, 97]. First it can be shown [98], that the mapping of S3 into a generic group G, which determines the topological classification of the instanton solutions, can be continuously deformed to a mapping into SU(2). Hence the classification of classical solutions into homotopy classes holds in general. Furthermore one can construct a solution in the case of a gauge group G by the embedding of a SU(2) instanton. In particular for G=SU(N) one can consider a configuration 2i ηa xν λ A = µν k , µ g x2(x2 + ρ2) 2

where λk, k =1, 2, 3, are the ﬁrst three generators of the fundamental representation

of SU(N). Under the SU(2) subgroup of SU(N) (λ1,λ2,λ3) transforms as a triplet, whereas the other generators are organized into 2(N 2) doublets and the remaining − are singlets. In conclusion, in addition to eight zero-modes of the SU(2) instanton, there are 4(N 2) additional zero-modes. The corresponding vectors take the form − [97]

1 λr x2 2 aˆr = D (A) . µ µ g x2 + ρ2 " # 4.2 Instanton calculus in the presence of fermionic ﬁelds

In this section the semiclassical quantization in the background of an instanton conﬁg- uration will be generalized to the case of a non Abelian gauge theory with interacting 4.2 Instantons and fermions 119

fermions. The Euclidean action of the model is

4 µ S[ψ, ψ, A]= SYM[A]+ SA[ψ, ψ]= SYM[A]+ d x ψ [iDµ(r)] σ ψ , (4.17) Z where SYM is the Yang–Mills action (4.3) and Dµ(r) denotes the covariant derivative in the representation r of the gauge group G (see appendix A for the notations). In (4.17) ψ is a Weyl fermion; an equivalent description can be given in terms of a Dirac spinor Ψ, introducing a ﬁctitious right chirality Weyl spinor and writing the action as

4 SA[Ψ, Ψ] = d x Ψ iD/ L(r) Ψ , Z where

µ 0 σ ∂µ D/ (r)= . L µ σ (∂µ + Aµ) 0 !

4.2.1 Semiclassical approximation

In the full interacting theory one is interested in computing expectation values of fields (x ,... ,x ) that are composite operators made up from the elementary fields ψ, ψ O 1 n and Aµ. In the semiclassical approximation one considers fluctuations, Aµ = Aµ +Qµ, around a classical solution Aµ; for the fermionic part of the action it is sufficient to restrict oneself to SA[ψ, ψ]. In the spirit of the semiclassical approximation, in the computation of expectation values of operators depending on Aµ as well as on fermionic fields, the functional integration over the latter is performed first and then the result is substituted into (4.14). Thus as a starting point one considers

SA[Ψ,Ψ] [Ψ, Ψ] = [ Ψ Ψ]e− [Ψ, Ψ] hO iA D D O Z and evaluates this integral through a saddle point approximation. According to the properties of Berezin integration (see appendix A) the last integral yields a non-vanishing result only if all the integrations over the fermionic variables are correctly saturated. Expanding Ψ and Ψ into eigenfunctions of the Dirac operator

(n) (n) (n) (n) iD/ L(r)f = λnf , iD/ R(r)g = λn∗ g , 120 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

gives

m (0) (0) (n) Ψ = bi fi + bnf i=1 n Xm X (0) (0) (n) Ψ = cj gj † + cng † , (4.18) j=1 n X X where b and c are Grassmannian c-numbers. Substituting into the action one gets

SA[Ψ, Ψ] = λncnbn , n=0 X6 so that one obtains m m (0) (0) λncnbn = [db ] [dc ] [db dc ] e− n=06 (b, c) . (4.19) hOiA i j n n O i=1 j=1 n=0 P Z Y Y Y6 This expression shows that the operator must contain exactly the number of zero- O modes of the Dirac operator that are present in the instanton background in order to have a non-vanishing vev. Hence one must study the eigenvalue problem for the Dirac operator in the instan-

ton background. The number, nL, of left-chirality zero-modes of the Dirac operator

(i.e. the number of zero-modes of iD/ L) and the number of right-chirality zero-modes

(i.e. the number of zero-modes of iD/ R), nR, are related by an index theorem [99]

n n =2drK[A] . (4.20) L − R

Moreover for conﬁgurations with self-dual (anti self-dual) ﬁeld strength one has nR=0

(nL=0). Thus from (4.19) in the K-instanton sector it immediately follows that the operator must contain an excess of exactly 2Kdr factors of Ψ with respect to Ψ to O yield a non-zero result. For instance in the case of a theory containing two Weyl fermions, ψ and χ, in the fundamental representation of the gauge group SU(2) the relation

nL =2drK[A] , nR = 0 when Fµν [A]= F˜µν [A] (4.21)

gives one single zero-mode, f (0) for ψ and χ in the K=1 sector, so that

(0) (n) ψ = a0f + anf n=0 X6 (0) (n) χ = b0f + bnf , n=0 X6 4.2 Instantons and fermions 121

(n) (n) while ψ = n=0 cng † and χ = n=0 dng † do not possess zero-modes. An opera- 6 6 tor with non-vanishing vev is (x, y) = ψα(x)χ (y). The saddle point semiclassical P O P α approximation to the expectation value of is O

λn(cnan+dnbn) = [da db ] [da db dc dd ]e− n=06 hOiA 0 0 n n n n · n=0 P Z Y6 ∞ ∞ (n) (n) 2 (0) (0) a f b f = det ′(iD/ (2)) f (x)f (y) , · n n L n=0 ! n=0 ! X X where det′ denotes the product of the non-zero eigenvalues. The generalization of this elementary example to more complicated cases is rather straightforward. In the following sections computations of vev’s of composite operators in =4 supersym- N metric Yang–Mills theory will be described. The =4 theory contains four Weyl N spinors in the adjoint representation. In the case of a SU(2) gauge group equation (4.21) with d3=2 yields sixteen zero-modes in the one-instanton sector.

4.2.2 Fermionic zero-modes

The discussion of the previous subsection shows that in order to compute the instanton contribution to expectation values of fermionic ﬁelds one needs to know the zero-value eigenstates of the Dirac operator. In particular their norms are necessary to convert the integrations over the zero-modes into integrations over (Grassmannian) collective coordinates as in the case of the bosonic zero-modes. For a Weyl spinor in the fundamental representation of SU(2) there is one single zero mode which satisﬁes the equation

(f) µ αα˙ (0) iDµ (A)σ ψα =0 ,

(f) where Dµ (A) denotes the Dirac operator in the fundamental of SU(2) in the K=1 instanton sector. The solution is ε ψ(0) (x)= αs , α,s [(x x )2ρ2]3/2 − 0 where s=1,2 is an index of the fundamental of SU(2). The norm of this zero eigen- function is π ψ(0) = . k k ρ 122 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

In the case of the adjoint representation the equation for the zero-modes becomes

iD(adj)(A) b σµ αα˙ λ(0) =0 . (4.22) µ a αb

This equation is more complicated and according to the Atiyah–Singer theorem [99] possesses four independent solutions in the case K=1 and G=SU(2). Instead of explicitly solving equation (4.22) the zero eigenfunctions can be obtained acting with the symmetries of the model 3 on the conﬁguration

λ = λ =0 , Aµ = Aµ ,

which trivially satisﬁes (4.22). More precisely the zero-modes are obtained acting by the symmetries broken by the instanton background. Two of the four zero-modes are generated by a supersymmetry transformation √ρ δ λ = F σµνη 1 2 µν and are associated with the supersymmetry parameter η. The remaining two eigenfunctions are obtained by a superconformal transformation with parameter ξ

1 µν κ δ2λ = F µν σ (x x0)κσ ξ . 2√ρ − The above four zero-modes can be written in the compact form 1 λ(0) = F σµν ζ (x) , (4.23) i 2 µν i where

(0) 1 µ ζi (x)= [ρηi +(x x0) σµξi 2] , i =1, 2, 3, 4 (4.24) √ρ − − and 1 0 η1 = ξ1 = , η2 = ξ2 = , η3,4 = ξ 1, 2 =0 . 0 1 − − (0) The powers of ρ in (4.24) are chosen in such a way as to give to ζi the correct dimen- 1 (0) sion, [mass]− 2 , for a supersymmetry parameter. The zero-modes λi are orthogonal and their norms are 4√2π√ρ 8π√ρ λ(0) = , j =1, 2 , λ(0) = , k =3, 4 . k j k g k k k g 3Note that this construction applies to non-supersymmetric cases as well. 4.3 Instanton calculus in SYM theories 123

The extension of the analysis described here to the case of a SU(N) gauge group, in the K=1 sector, can be obtained without too much effort and is achieved by the embedding of a SU(2) instanton in the SU(N) field configuration, see [100]. The generalization to multi-instanton sectors on the contrary is quite involved and has been developed in [101, 102, 103, 104, 105].

4.3 Instanton calculus in supersymmetric gauge theories

The machinery of the previous sections allows to compute the instanton contribution to the vacuum expectation value of composite operators in supersymmetric gauge theories. The problem is reduced to the computation of the “primed” determinants of the kinetic operators, i.e. the product of non-zero eigenvalues, and then to the integration over the collective coordinates with the measure constructed in sections 4.1 and 4.2. In non-supersymmetric theories the integration over the bosonic collective coordinates is IR-divergent because of the singular behaviour at ρ 0. In supersymmetric → gauge theories, on the contrary, in all the known cases the integration turns out to be finite thanks to the balance of fermionic and bosonic degrees of freedom that controls the ρ-dependence. This exact balance produces another significant simplification: the product of non-zero eigenvalues of bosonic and fermionic kinetic operators exactly cancels out. As a result in various models instanton calculus allows to obtain finite and exact results (in the semiclassical approximation) for the vev’s of composite operators, see [87] for a review.

Cancellation of determinants in supersymmetric theories.

Supersymmetry implies relations among the eigenvalues of the kinetic operators for scalars, vectors and spinors, that lead to an exact compensation in the ratio of determinants that enters the expression of the expectation values in the semiclassical approximation.

First notice that the chiral Dirac operator iD/ L(r) satisﬁes

2 det iD/ (r) = det iD/ (r)iD/ (r) = det iD/(r) det( 2) , L L R −

µ where iD/(r ) = iγ Dµ(r ) is the kinetic operator for a Dirac spinor, so it is suﬃcient 124 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

to solve the eigenvalue problem for iD/(r). Let λn be the non zero solutions of the eigenvalue problem for a scalar ﬁeld in the instanton background

D2(A)ϕ(n) = λ2 ϕ(n) λ =0 . − n n 6 The two eigenvalue problems to be solved are

(n) (n) iD/(A)Ψ = µnΨ D2(A)δ 2F [A] Q(n) = ρ2 Q(n) − µν − µν ν n µ and one can prove the following relations

ξ(n) µ = λ Ψ(n)(x)= , n n ←→ χ(n) with ξ(n) = i D (A)ϕ(n)σµǫ, χ(n) = ǫϕ(n) and λn µ

ρ = λ Q(n) = ησ ξ(n) , n n ←→ µ µ where η and ǫ are two constant Weyl spinors. The ratio of determinants that enters the expression for the vev of an operator in the semiclassical approximation is

1 (adj) g.f. 2 2 2 / det(M + µ ) det D (A) det′ iDL µν , g.f. 2 2 (adj) det′ Mµν det(D (A)+ µ ) " # det iD/ L + iµ where a regularization `ala Pauli–Villars has been introduced. Now substituting the eigenvalues found in the previous sections one can show that there is a an exact cancellation between bosons and fermions, so that the contribution of the non-zero modes is 1! The total contribution of the zero-modes is

1 nB nF µ − 2 , (4.25)

where nB and nF are respectively the number of bosonic and fermionic zero-modes. Notice that in the case of =4 super Yang–Mills theory (4.25) reduces to 1, as is N expected, since the theory, being finite, cannot lead to a result depending on the renormalization scale. For the calculation of vev’s of operators there are two different situations to O be considered. If the operator contains precisely the number of zero-modes to O 4.4 Instanton calculations in =4 SYM theory 125 N saturate the fermionic integrations in the instanton background, one must simply substitute the fermionic fields in with the zero-modes and perform the integration O over the collective coordinates. This is the case that will be discussed in section 4.6. If instead the operator does not contain a sufficient number of fermionic insertions O to saturate the Grassmannian integrals, then the correlator is zero at the lowest order and one must consider in the insertion of terms lowered from the action, until hOi the correct number of spinors is obtained. This case will be described in section 4.5 for the correlation function of four scalar fields.

4.4 Instanton calculations in =4 supersymmet- N ric Yang–Mills theory

The formalism described in the preceding sections will now be applied to the computation of Green functions of gauge invariant composite operators in =4 supersym- N metric Yang–Mills theory. In particular correlation functions of operators belonging to the multiplet of currents, see equations (2.17) and (2.18), will be calculated to lowest order in the one-instanton sector and with gauge group SU(2) [89]. The extension of the results presented here to the case of SU(N) and to the K-instanton sector in the large N limit was given in [90] and [91, 92] and will be briefly reviewed in the concluding section. The operators in the current multiplet that will be considered play a central rôle in the correspondence with type IIB superstring theory compactified on AdS S5. This issue will be discussed in chapter 5, where the instanton 5 × calculations will be compared with D-instanton effects in the type IIB superstring on AdS S5. 5 × In [89] a closed form for the instanton contribution to a four-point function of composite scalar operators, ij in the 20 of the SU(4) R-symmetry group has been Q given. In the next section the complete space-time dependence of this Green function will be computed. As will be discussed it will prove useful to employ for this calculation a description of the model in terms of =2 multiplets. Also, it will be N shown that when a limit of coincident points is considered, in order to study the operator product expansion (OPE), the function develops a logarithmic singularity. Analogous results have been found by various groups [106, 107, 108, 109]. In these papers a similar behaviour was extracted without explicitly performing all the integrations, on the contrary in the computation presented here the singular behaviour is shown with no ambiguities after a complete evaluation of the integrals which yield 126 Chapter 4. Non-perturbative effects in =4 SYM theory N

the Green function under consideration in the semiclassical approximation. A more detailed analysis of this subject and related questions will be presented elsewhere [110]. In section 4.6 the computation of correlation functions of sixteen fermionic currents Λˆ A and of eight gaugino bilinears AB will be reported. For these Green func- α E tions the integration over the collective coordinates will not be performed, nevertheless the unintegrated form presented here will be suﬃcient for the comparison with the results from type IIB superstring theory discussed in the ﬁnal chapter.

4.5 A correlation function of four scalar supercurrents

The multiplet of superconformal currents of =4 super Yang–Mills theory has been N given in section 2.2 for the Abelian case. The natural four-point function of superconformal currents to consider would be a correlator of four stress energy tensors. However, due to its complicated tensorial structure even the free-ﬁeld expression for this correlator is awkward to express compactly and it is much simpler to consider correlations of four gauge-invariant composite scalar operators 1 ij = ϕiϕj δijϕ ϕk , i, j, k =1, 2,..., 6 , (4.26) Q − 6 k belonging to the representation 20 of the SU(4) R-symmetry group. As discussed in chapter 2 ij is the lowest component of the composite twisted chiral current Q superﬁeld [111] ij W(2) δij ij = tr W iW j W W k . W(2) − 6 k After calculating correlation functions of these components one can derive those of any other component in the = 4 supercurrent multiplet by making use of the N superconformal symmetry [111]. A way to explicitly do it may be to resort to analytic superspace, recalling that

ij ij (x) (2)(Υ) , Q ∼ W θ=θ¯=0

where Υ are the supercoordinates of analytic superspace [21]. Therefore, by computing correlators of ij(x) and substituting x Υ any of the other correlation Q → 4.5 Instanton contribution to a four-point function 127

functions can in principle be obtained by expanding in the fermionic as well as in the auxiliary bosonic coordinates of analytic superspace. The correlation function that will be considered is therefore

i1j1 (x ) i2j2 (x ) i3j3 (x ) i4j4 (x ) . (4.27) hQ 1 Q 2 Q 3 Q 4 i The value of this correlation function in the free ﬁeld theory is determined from the expression for the free two-point scalar Green function which is

1 δijδab ϕia(x)ϕjb(y) = . (4.28) h ifree (2π)2 (x y)2 − Hence, the free-ﬁeld expression for the correlation function that follows by Wick contractions is

i1j1 (x ) i2j2 (x ) i3j3 (x ) i4j4 (x ) (4.29) hQ 1 Q 2 Q 3 Q 4 ifree 1 δi1i3 δj1j3 δi2i4 δj2j4 δj4i1 δj1i3 δj3i2 δj2i4 = N 4 + N 2 + permutations , (4π2)4 x4 x4 x2 x2 x2 x2 13 24 41 13 32 24 where,

x = x x . ij i − j The ﬁrst term in this expression is simply the product of two two-point functions and is known to be exact. The second term, which is the free-ﬁeld contribution to the connected four-point function, certainly gets corrections from interactions. As has been discussed in detail in the preceding sections, in =4 Yang–Mills N theory there are sixteen zero-modes in the one-instanton sector in the case at hand of a SU(2) gauge group. In the background of the one-instanton solution, (4.9), (4.10), the correlation function

i1j1 (x ) i2j2 (x ) i3j3 (x ) i4j4 (x ) (4.30) hQ 1 Q 2 Q 3 Q 4 iK=1 where the subscript K = 1 denotes the winding number of the background, is zero at the lowest order, because it does not contain any fermionic insertion to saturate the integrations over the Grassmannian collective coordinates of the sixteen zero- modes. However it is easy to check that the correlation function (4.30) soaks up these sixteen gaugino zero-modes when the perturbative corrections to the standard instanton conﬁguration are considered. This can be seen either from the form of the 128 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

supersymmetry transformations, see equation (2.6), or from the equation of motion for the scalar fields ϕi. The expression (4.10) is annihilated by the conserved super- α˙ symmetry transformations (those associated with the parameter η A in (2.6)), while A the transformations corresponding to supersymmetries associated with ηα are broken and generate eight of the sixteen fermionic zero modes. The other eight zero-modes are generated by superconformal transformations. As a consequence acting with the broken supersymmetry and superconformal transformations on the scalar fields in (4.30) produces a configuration which possesses the required sixteen zero-modes. To obtain the first non-vanishing contribution to (4.30) one must consider insertions of terms lowered from the action until the correct number of fermionic fields is obtained. The first non vanishing correction comes from the insertion of a Yukawa term for each field ϕi (see the form (2.5) of the action)

i i i1j1 (x ) i2j2 (x ) i3j3 (x ) i4j4 (x ) d4z gf abct λαAλB ϕ (z ) hQ 1 Q 2 Q 3 Q 4 1 4 AB a αb ic 1 · Z i . . . d4z gf abcti λαAλB ϕ (z ) . (4.31) · 8 4 AB a αb ic 8 iK Z Wick contractions among the scalar ﬁelds produce propagators, so that (4.31) becomes schematically

d4z [g∆(x z )(λλ)(z )] . . . d4z [g∆(x z )(λλ)(z )] , (4.32) h 1 1 − 1 1 8 4 − 8 8 i Z Z where for simplicity of notation not all the indices have been indicated explicitly. Computing the integrations in (4.32) amounts to substitute each factor with the (0) solution, ϕi , of an equation of the form

D2(A)ϕi (x)= J i(x) , (4.33)

i i A B with ‘source’ J (x)= tAB λ λ (x). In order to evaluate this instanton contribution in the most convenient fashion it will be convenient to use the =2 supersymmetric description in which, as explained N in section (2.1), ϕi decomposes into the complex =2 singlet ϕ with U(1) charge +2 N and a neutral =2 ‘quaternion’ qT in the (2, 2)0 representation of SU(2) SU(2) N V × H which resides in the =2 hypermultiplet. The scalar component AB of the =4 N Q CD N current decomposes in the following way in terms of =2 ﬁelds, N S0 = qSϕ S0† = qSϕ ST = qSqT trace Q Q Q − 2 2 (+) = ϕ ( ) = ϕ (0) = ϕϕ trace , (4.34) Q Q − Q − 4.5 Instanton contribution to a four-point function 129

S S where the notations are those of section (2.1) and quu˙ = qS σ uu˙ and σ = (1I, ~σ), with ~σ the standard Pauli matrices (and S, T =1,..., 4 are SU(2) SU(2) vector V × H indices). The correlation function of two ϕ2 and two ϕ2 will be discussed here. From (4.30) the free-ﬁeld expression for this particular correlator is

ϕ2(x )ϕ2(x )ϕ2(x )ϕ2(x ) = h 1 2 3 4 ifree 1 4N 4 4N 4 16N 2 = + + , (4.35) (4π2)4 x4 x4 x4 x4 x2 x2 x2 x2 13 24 14 23 41 13 32 24 where in the previous notation ϕ2 = 55 66 +2i 56. Q(+) ≡ Q − Q Q The =2 formalism is particularly suitable to perform the integrations over the N fermionic zero-modes because it follows from the structure of the Yukawa couplings (2.13) that ϕ only absorbs those zero modes of the =4 gauginos that belong to the N =2 vector multiplet (λu), while ϕ absorbs the zero modes belonging to the =2 N α N u˙ hypermultiplet (ψα). The expressions for the zero-modes that will be used are those suggested by the supersymmetry transformations of the instanton

u 1 µν β 1 u κ ¯βu˙ λ(0)α = F µν σα ρ0ηβ +(x x0)κσββ˙ ξ . (4.36) 2 √ρ0 −

u˙ and similarly for ψ(0)α. Here and in the following the ‘size’ of the instanton will be denoted by ρ0. The reason for this choice of notation will become clear when the results of this chapter will be reanalyzed in the context of the AdS/CFT correspon- β˙ dence in chapter 5. In this decomposition ηβ and xββ˙ ξ are the parameters of the broken supersymmetry and special supersymmetry transformations respectively. One u can assemble the fermionic collective coordinates into the spinors ζ (ρ0, x) (where ± ± refers to the U(1) R-symmetry charge) to parameterize the fermionic zero modes

u 1 u µ αu˙ ζ α(ρ0, x x0)= ρ0η α + σ αα˙ (xµ x0 µ)ξ , (4.37) ± − √ρ0 ± − ±

Using the bosonic measure for SU(2) instantons derived in section 4.1 and the norms of the fermionic zero-modes given in section 4.2 and proceeding as described after equation (4.30) one ﬁnds that in the semi-classical approximation the one-instanton 130 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

contribution to the four bosonic current Green function is 4

8 8π2 2 2 2 2 2 2 2 2 g 2 +iθ 4 − g G (xp)= g ϕ (x1)g ϕ (x2)g ϕ (x3)g ϕ (x4) K=1 = e Q h i 232π10 4 dρ0d x0 4 4 4 4 2 2 2 2 5 d η+d ξ+d η d ξ ϕ(0)(x1)ϕ(0)(x2)ϕ(0)(x3)ϕ(0)(x4) , (4.38) ρ − − Z 0 where (+) (or ( )) refers to the U(1) charges of the gauginos in the vector (or hyper) − multiplet (see below). In equation (4.38) the explicit dependence on the vacuum

angle, θ, has been indicated. The Green function G 4 receives a contribution also Q from the K = 1 sector that is the complex conjugate of (4.38). − In (4.38) the ﬁelds ϕ and ϕ have been replaced by the expressions

1 u µν v ϕ(x) ϕ(0)(x)= εuvζ σ ζ F µν → 2√2 + + 1 u˙ µν v˙ ϕ(x) ϕ(0)(x)= εu˙v˙ ζ σ ζ F µν , (4.39) → 2√2 − − which satisfy an equation of the form of (4.33) and are the leading nonvanishing terms that result from Wick contractions in which Yukawa couplings are lowered from the exponential of the action until a suﬃcient number of fermion ﬁelds are present to saturate the fermionic integrals. Of course, these expressions can also be obtained

directly from the supersymmetry transformations (2.6) by acting twice on F µν with the broken supersymmetry generators. After some elementary Fierz transformations on the fermionic collective coordinates the fermionic integrations can be performed in a standard manner and the result is

4 2 4 4 4 3 8π +iθ dρ0d x0 ρ0 8 g2 4 4 G 4 (xp)= g e− x12x34 . (4.40) Q 4π10 ρ5 ρ2 +(x x )2 0 p=1 0 p 0 Z Y − The integration in (4.40) resembles that of a standard Feynman diagram with momenta replaced by position diﬀerences and can be performed by introducing the Feynman parameterization,

4 2 3 Γ(16) 8π +iθ 8 g2 3 G 4 (xp) = g e− αpdαp δ 1 αq Q 4π10 (Γ(4))4 − · Z p q ! 4 4 Y4 16 X dρ0d x0 x12x34 ρ0 5 16 . (4.41) · ρ0 2 2 2 Z ρ + x 2x0 αpxp + x 0 0 − · p p p 4 Here and in the following computations of correlatorsP of currentP bilinears in the fundamental Yang–Mills fields a factor of g2 is included for each external insertion. It would be equivalent to rescale the fundamental Yang–Mills fields according to A A′ = gA and compute the Green functions of primed fields. This leads to a common overall dependence→ on g2 for the three correlation functions that will be considered. 4.5 Instanton contribution to a four-point function 131

The ﬁve-dimensional integral yields

3 8π2 3 Γ(11) 8 2 +iθ G 4 (xp) = g e− g Q 27 (π3Γ(4))4 · 4 4 3 x12x34 αpdαp δ 1 αq . (4.42) · − q 8 p α α x2 Z Y X p p q pq This integral can be simpliﬁed by observing that it is essentP ially obtained by acting with derivatives on the box-integral with four massless external particles,

3 2 3 Γ(11) 8π +iθ ∂ 8 g2 4 4 G 4 (xp)= g e− x12x34 B(xpq) , (4.43) Q 7 3 4 ∂x2 2 (π Γ(4)) p

4 1 u+u u+ − B(xpq) = log 2 2 log + ∆(xpq) 2 (1 u+) (1 u ) u − − − − 1 1 Li2(1 u+p)+Li2(1 u ) Li2 1 + Li2 1 , (4.45) − − − − − − u − u+ − where

2 2 2 2 ∆ = det((xpq)) = X + Y + Z 2XY 2YZ 2ZX (4.46) 4 4 × − − − and Y + X Z √∆ u = − ± , (4.47) ± 2Y 2 2 2 2 2 2 with X = x12x34, Y = x13x24 and Z = x14x23.

Notice that up to an overall dimensional factor needed for the correct scaling, G 4 Q turns out to be a function of the two independent superconformally invariant cross ratios X/Z and Y/Z. Although not immediately apparent, the expression B(xpq) is symmetric under any permutation of the external legs, as can be seen by making use of the properties of the dilogarithms, π2 Li (z)+Li (1 z) = log(z) log(1 z) 2 2 − 6 − − 1 π2 1 Li (z)+Li = [log( z)]2 , (4.48) 2 2 z − 6 − 2 − 132 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

and observing that the relevant permutations correspond to permutations of X =

Yu+u , Y and Z = Y (1 u+)(1 u ), that are generated by the two transformations: − − − − a) u+ 1/u , u 1/u+, Y Yu u+, which is equivalent to the exchange of X → − − → → − and Y , leaving Z ﬁxed (i.e. to the exchange of x1 and x4 or, equivalently, of x2 and

x3) and b) u+ 1 u , u 1 u+ at ﬁxed Y , which is equivalent to the exchange → − − − → − of X and Z (or the exchange of x2 and x4 or, equivalently, of x1 and x3).

To compute the explicit dependence of G 4 on xp one must perform the six deriva- Q tives with respect to xpq. These can be more conveniently calculated rewriting the diﬀerential operator in (4.43) as ∂ = , (4.49) ∂x2 DZ DY DX p

11X8 28X7Y 52X6Y 2 + 284X5Y 3 430X4Y 4 + 284X3Y 5 52X2Y 6 + − − − − − 28XY 7 + 11Y 8 + 62X7Z + 590X6YZ 2142X5Y 2Z + 1490X4Y 3Z + − − +1490X3Y 4Z 2142X2Y 5Z + 590XY 6Z + 62Y 7Z 331X6Z2 + 972X5YZ2 + − − +4491X4Y 2Z2 10264X3Y 3Z2 + 4491X2Y 4Z2 + 972XY 5Z2 331Y 6Z2 + − − +362X5Z3 4894X4YZ3 + 5132X3Y 2Z3 + 5132X2Y 3Z3 4894XY 4Z3 + − − +362Y 5Z3 + 215X4Z4 + 3404X3YZ4 8982X2Y 2Z4 + 3404XY 3Z4 + − +215Y 4Z4 646X3Z5 + 1170X2YZ5 + 1170XY 2Z5 646Y 3Z5 + 383X2Z6 + − − XY 1180XYZ6 + 383Y 2Z6 34XZ7 34YZ7 22Z8 log + − − − − Z2 36 + X9 +3X8(Y + Z) 6X7 5Y 2 19YZ +5Z2 +(Y Z)6 Y 3 +9Y 2Z + ∆13/2 − − − +9YZ2 + Z3 + X6 62Y 3 144Y 2Z 144YZ2 + 62Z3 +3X(Y Z)4 − − − · Y 4 + 42Y 3Z + 114Y 2Z2 + 42YZ3 + Z4 6X5 6Y 4 + 83Y 3Z 252Y 2Z2 + · − − 2 +83 YZ3 +6Z4 6X2(Y Z) 5Y 5 + 34 Y 4Z 189Y 3Z2 189Y 2Z3 + 34YZ4 + − − − − +5Z5 6X4 6Y 5 175Y 4Z + 223 Y 3Z2 + 223Y 2Z3 175YZ4 +6Z5 + − − − +X3 62Y 6 498Y 5Z 1338Y 4Z2 + 3948Y 3Z3 1338Y 2Z4 498YZ5+ 62Z6 − − − − · 1 XY (X + Y Z)√∆ + ∆ X Y+Z √∆ log log 1+ − Li − − + · 2 Z2 2XY − 2 2X X+Y+Z √∆ X Y+Z+ √∆ X+Y+Z+ √∆ Li − − + Li − + Li − . − 2 2Y 2 2X 2 2Y In conclusion the one-instanton contribution to the correlation function G 4 (xp) is Q 3 2 3 Γ(11) 8π +iθ 8 g2 4 4 G 4 (xp)= g e− x12x34 Q(xpq) . (4.50) Q 25 (π3Γ(4))4 Unlike correlation functions of elementary fields that are infra-red problematic and gauge-dependent, the above correlator is well defined at non-coincident points. Moreover it possesses the correct symmetry properties. This can be derived from the symmetry of B(xpq) discussed above and the observation that the differential operator (4.49) is completely symmetric. Of course the symmetry properties can be checked directly on the final expression for Q(xpq). Notice that the above form for

Q(xpq) is valid only for X,Y,Z in the ‘physical domain’, i.e. in the region obtainable 2 for allowed choices of xpq, that is deﬁned by the condition ∆=∆(X,Y,Z)= X2 + Y 2 + Z2 2XY 2XZ 2YZ 0 . − − − ≤ 134 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

In this region single terms in the above expression are complex, but the complete

function is real. To describe the physical region it is convenient to rewrite Q(xpq) in terms of two independent cross ratios, 2 2 2 2 X x12x34 Z x14x23 r = = 2 2 , s = = 2 2 , Y x13x24 Y x13x24 1 extracting a factor Y 4 in Q(xpq). Then the physical domain corresponds to the region ∆ δ(r, s)= =1+ r2 + s2 2r 2s 2rs 0 Y 2 − − − ≤ in the (r, s) plane. This is the region inside the parabola in ﬁgure 4.1

s δ=1-2r-2s-2rs+r +s2 2

δ<0

0 1 r

Figure 4.1: Physical domain for B(xpq)

Having obtained a closed form for G 4 (xp) one can study the behaviour when Q any two points are taken close to one another, x 0. For instance x 0 corre- pq → 12 → sponds to r 0, s 1 simultaneously. In this limit B(x ) develops a logarithmic → → pq singularity.

The logarithmic divergence in the Green function 4 can be shown more clearly GQ considering the particular conﬁguration in which the points xp, p = 1, 2, 3, 4, are taken on a line. In this case the cross ratios r and s are not independent. Assuming

x1 < x2 < x3 < x4 r and s are related by √r + √s =1 . 4.5 Instanton contribution to a four-point function 135

Thus one can express the result in terms of a single variable η deﬁned by

r = η2 , s = (1 η)2 . (4.51) −

5 In this limit the above computed function Q(xpq) reduces to

log(η2) Q˜ = Q˜(η)=3 (100 η6 + 429+ 2431 η2 2717 η3 + 1794 η4 + ( 1+ η)7 − − log[( 1+ η)2] 650 η5 1287 η) 3 − (100 η6 + 50 η5 + 44 η4 + 41 η3 + − − − η7 (η2 η + 1)2 +44 η2 + 50 η + 100) 2 − (300 900 η + 307 η2 + 886 η3 + − η6 ( 1+ η)6 − − +307 η4 900 η5 + 300 η6) . (4.52) − and it can be shown that Q˜(η) has a logarithmic singularity as η 0 or η 1. → → This peculiar behaviour has been observed by various authors in four-point functions computed in the context of the AdS/CFT correspondence [106, 107, 108, 109], but here it is derived from an exact field theoretical computation and thus shown with no ambiguities. Notice that single terms in Q(xpq) have pole-type singularities in the limit x 0, but in the sum the singularity is only logarithmic. Note in 12 → particular that the poles associated with operators of dimension lower or equal to 2 in the OPE are absent. Moreover with non-vanishing Y the only singularities in Q B(xpq) correspond to the points (r = 0,s = 1) and (r = 1,s = 0) marked in figure 4 4 4.1. Taking into account the pre-factor, x12x34, in the complete expression of G 4 one Q finds that the limit in which two ϕ2 operators are taken to coincide gives a vanishing result, while a logarithmic singularity is actually present when ϕ2 approaches ϕ2. The logarithms by themselves do not violate the superconformal symmetry of the theory, that is manifest in the final expression of the four-point function. In some respect, logarithmic behaviours in four-point functions should not sound unexpected in a superconformal theory, such as =4 supersymmetric Yang–Mills theory, that N contains a large number of primary fields all of whose (protected) dimensions are integer. The singularity may be due to the presence of an infinite tower of stable BPS dyons that collapse to vanishing mass and size in the superconformal phase or to a gas of instantons of small size from the Euclidean viewpoint. This problem is under active investigation [110].

5This expression has been obtained by Yassen Stanev starting from a diﬀerent but equivalent formula for Q(xpq). 136 Chapter 4. Non-perturbative eﬀects in =4 SYM theory N

4.6 Eight- and sixteen-point correlation functions of current bilinears

In this section the one-instanton contribution to correlation functions of sixteen fermionic bilinears Λˆ A and of eight gaugino bilinears AB in the current supermultiplet α E will be calculated in the semiclassical approximation. The expressions obtained will be reconsidered in the next chapter and a comparison will be made with D-instanton eﬀects in type IIB superstring theory compactiﬁed on AdS S5. 5 × 4.6.1 The correlation function of sixteen fermionic currents

As could have been anticipated, it is particularly simple to analyze the contribution of the Yang–Mills instanton to the correlation function of sixteen of the fermionic ˆ A µν β A superconformal current bilinears, Λα = tr σ α Fµν− λβ ,

16 2 ˆ Ap Gˆ 16 (x )= g Λ (x ) , (4.53) Λ p h αp p iK=1 p=1 Y Since each factor of Λˆ in the product provides a single fermion zero mode it is necessary to consider the product of sixteen currents in order to saturate the sixteen

Grassmannian integrals. To leading order in g, GΛˆ 16 does not receive contribution from anti-instantons. The leading term in the one-instanton sector is simply obtained A by replacing each Fµν− with the instanton proﬁle F µν (equation (4.10)) and each λα A with the corresponding zero mode, λ(0)α

1 1 βA˙ λA = F σµν β ρ ηA +(x x ) σµ ξ . (4.54) (0)α µν α 0 β 0 µ ββ˙ 2 √ρ0 − The resulting correlation function thus has the form

62 16 2 4 2 3 8π +iθ d x0 dρ0 8 g2 8 8 Gˆ 16 (x )= g e− d ηd ξ Λ p π10 ρ5 · Z 0 Z 16 4 ρ 1 α˙ pAp 0 ρ ηAp +(x x ) σµ ξ . (4.55) · [ρ2 +(x x )2]4 ρ 0 αp p − 0 µ αpα˙ p p=1 0 p 0 √ 0 Y − The integration over the fermion zero modes leads to a sixteen-index invariant ten-

sor, t16, of the product of the SU(4) and Lorentz groups. Assembling the 16 fer-

mionic collective coordinates into a sixteen-dimensional spinor t16 would simply read

a1a2...a16 a1a2...a16 t16 = ε , with ai = 1, 2 . . . 16. Further integration over the instanton 4.6 Instanton contribution to eight- and sixteen-point functions 137

moduli space would determine the dependence on the coordinates xp. However, for the purpose of the comparison with the corresponding expression obtained in the type IIB string theory in AdS S5, that will be discussed in the next chapter, it is 5 × suﬃcient to leave the expression in the unintegrated form (4.55).

4.6.2 The correlation function of eight gaugino bilinears

In a similar fashion it is easy to deduce the one-instanton contribution to other related processes, such as the eight-point correlation function,

2 A1B1 2 A8B8 8 G (xp)= g (x1) ...g (x8) K=1 , (4.56) E h E E i which also saturates the sixteen fermionic zero-modes present in the SU(2) one- instanton background. To leading order in g, G 8 does not receive contribution from E anti-instantons. The complete non-abelian expression for AB was given in equation E (2.19) and reads

AB = λαaAλ B + gf t(AB)+ φiaφjbφkc , (4.57) E αa abc ijk but at leading order in the gauge coupling constant only the term proportional to the gaugino bilinear is relevant. In the instanton background the gaugino bilinear is given by

6 4 αaA B 3 2 ρ0 αA B λ λ(0)αa = · ζ ζα . (4.58) (0) g2 (ρ2 +(x x )2)4 0 − 0 Then it follows

8 14 2 4 3 2 8π +iθ d x0 dρ0 8 g2 8 8 G 8 (xp)= g e− d ηd ξ E π10 ρ5 · Z 0 Z 8 4 ρ0 1 Ap µ Apα˙ p ρ0η +(xp x0) σ ξ · 2 2 4 ρ αp − µ αpα˙ p · p=1 "(ρ0 +(xp x0) ) √ 0 Y − 1 Bpβ˙p εαpβp ρ ηBp +(x x ) σν ξ . (4.59) 0 βp p 0 ν βpβ˙p · √ρ0 − The integration over the fermion zero modes leads to an SU(4) invariant contraction of the sixteen-index tensor t16 defined after (4.55) and further integration over the instanton moduli space would determine the exact dependence on the coordinates xp. Again the unintegrated expression (4.59) is sufficient for comparison with the D-instanton contribution to the corresponding AdS S5 amplitude that will be 5 × considered in the next chapter. 138 Chapter 4. Non-perturbative effects in =4 SYM theory N

4.7 Generalization to arbitrary N and to any K for large N

In the previous sections instanton calculus in =4 supersymmetric Yang–Mills theory N with gauge group SU(2) in the K=1 sector have been presented following [89]. These results have been extended to the case of SU(N) for generic N in [90] and to arbitrary instanton number in the large N limit in [91, 92]. These extensions are of fundamental importance for the re-interpretation of the results of this chapter in the context of the AdS/CFT correspondence that will be discussed in chapter 5. In this section the main steps of the extensions of [90, 91, 92] will be briefly reviewed. The generalization to N > 2 is rather straightforward, on the contrary instanton calculus in sectors K > 1 presents an enormous increase of computational complexity. The formalism for multi- instanton calculations (ADHM formalism) has been developed in [101, 102, 103, 104, 105] and a review of the application of these techniques to SU(N) supersymmetric gauge theories can be found in [113]. As has been explained in [91, 92] there are remarkable simplifications in the ADHM formalism in the large N limit. Since for a gauge group SU(N) there are 2N gaugino zero modes in the · N one-instanton background it might appear from a superficial analysis that the Λˆ 16- correlation function should vanish for N > 2. However it has been argued in [89] and then clearly shown in [90] that only the sixteen ‘geometric’ (8 supersymmetric + 8 superconformal) zero-modes are actually relevant in the general case of SU(N), since A A all the remaining ones, µi , µi , are ‘lifted’ by a four-fermion term that is generated in the one-instanton action. The instanton action computed in [90] has the form 8π2K S = + S , (4.60) inst g2 4F where S contains a quartic fermionic interaction built up with the 8(N 2) ad- 4F − ditional fermionic collective coordinates. The exact form of S4F in the K=1 sector is

N 2 N 2 π2 − − S = ε µAµB µC µD . (4.61) 4F 16ρ2g2 ABCD i i j j " i=1 # " j=1 # X X

S4F is supersymmetric and lifts all but the 16 ‘geometric’ gaugino zero modes. For deﬁniteness in the following the attention will be focused mainly on the Λˆ 16 correlator, that is studied in [90]. In the case of SU(N) there are additional zero- A modes contributing to λ(0)α in (4.54). However these are lifted by (4.61) and the 4.7 Generalization to any N and to K > 1 for large N 139

lowest order contribution in the one-instanton background is still obtained by substituting (4.54) in each factor of Λ.ˆ As a result the expression for the Λˆ 16-correlator reviewed above for the case N=2 is actually true for all N, apart from an overall N-dependent factor

4 N 2 8π2 4 − N 8 2 +iθ d x0 dρ0 8 8 A A S G (x )= C g e− g d ηd ξ dµ dµ e− 4F Λˆ 16 p N ρ5 i i · 0 A=1 i=1 Z Z Z Y Y 16 4 ρ 1 α˙ pAp 0 ρ ηAp +(x x ) σµ ξ , (4.62) · [ρ2 +(x x )2]4 ρ 0 αp p − 0 µ αpα˙ p p=1 0 p 0 √ 0 Y − where CN is a N-dependent constant determined by the norms of the bosonic and fermionic zero-modes. To completely determine the dependence on N one must still A A compute the integrals over µi and µi

4 N 2 − A A S4F IN = dµi dµi e− . (4.63) A=1 i=1 Z Y Y In computing the last integral it is convenient to perform a Hubbard-Stratonovich transformation of the fermion bilinears and represent S4F as a Gaussian integral i over auxiliary bosonic ‘collective coordinates’ χAB [90]. In conclusion one ﬁnds the following overall N dependence [90]

N G (x ) √NGˆ 16 (x ) , (4.64) Λˆ 16 p ∼ Λ p √ where GΛˆ 16 (xp) is the expression valid in the N=2 case. This factor of N will be of relevance in the next chapter for the comparison with D-instanton effects in type IIB superstring theory. The generalization to multi-instanton sectors in the large N limit, as derived in [91], is much more involved and would require an extended discussion, so only the basic ideas will be summarized here. A detailed report on the construction can be found in [92]. The moduli space of a SU(N) K-instanton configuration, known as ADHM moduli space, has a very complicated structure for generic N. In [91, 92] it has been proved that in the large N limit it is dominated by K instantons living in K mutually orthogonal SU(2) subgroups of SU(N). Then a saddle point approximation is used to show that the geometry is actually described by (AdS S5)K, where in each 5× i i factor AdS5 is parameterized by the positions and sizes of the K instantons, (x0 µ, ρ ), 5 i i = 1,...,K, and S by the auxiliary bosonic coordinates, χAB, introduced in the 140 Chapter 4. Non-perturbative effects in =4 SYM theory N

Hubbard–Stratonovich transformation. Moreover the integrations in the vicinity of the large N saddle point generate an attractive potential which actually forces all the K instantons to a single point reducing the moduli space to a single copy of AdS S5 [92]. Finally the small ﬂuctuations about this reduced moduli space 5 × describe the dimensional reduction of SU(K) =1 super Yang–Mills from d=10 to N d=0+0, so that the K-instanton measure in the large N limit factorizes into the product of the measure on AdS S5 times the partition function for the SU(K) 5 × ZK (0+0)-dimensional theory. In conclusion, after evaluating , see [114, 115, 116, 117], ZK one obtains for a n-point correlation function like those considered in this chapter an expression of the form [92]

2 7 8π K (K) 8 n 2 1 G (x )= g √NK − 2 e− g F (x ) , (4.65) n p d2 n p d K X|

where the sum is over the divisors of K and the function Fn(xp) is independent of K. Chapter 5

AdS/SCFT correspondence

The remarkable proposal suggested by Maldacena in [38] of a correspondence relating type IIB superstring theory in AdS S5 to =4 super Yang–Mills theory has led to a 5 × N great renewal of interest in the latter. According to the formalization of Maldacena’s original idea given by Gubser, Klebanov and Polyakov [118] and by Witten [119] =4 supersymmetric Yang–Mills theory in the superconformal phase is dual to type N IIB superstring theory in AdS S5 in the sense that correlation functions of Yang– 5 × Mills gauge invariant composite operators can be derived from the computation of amplitudes in the compactification of the type IIB superstring on AdS S5. The 5 × precise interpretation of this duality will be discussed later. More generally the main idea of [38] is that type IIB superstring theory in AdS M, where M is a compact manifold with positive curvature, should be d+1 × equivalent to a (super) conformal field theory living on the d-dimensional boundary of AdSd+1. In this respect the proposed correspondence is holographic in the sense of [120, 121]. Moreover in the case of superconformal field theories with gauge group SU(N) Maldacena has argued that the large N limit should be equivalent to the supergravity approximation to the AdS compactification of the type IIB superstring. This consideration makes the proposed duality particularly intriguing because it has long been believed that the large N behaviour should be crucial in the understanding of the non-perturbative dynamics of non Abelian gauge theories [122]. Non trivial checks of the suggested correspondence have been obtained in the large

141 142 Chapter 5. AdS/SCFT correspondence

N limit, i.e. employing the supergravity approximation, both at the perturbative and at the non-perturbative level. However very little has been said about the extension to finite N and the rôle of string corrections, although arguments have been developed supporting the validity of the correspondence for finite N as well. The computation of four-point functions in the context of the AdS/SCFT correspondence has also allowed to point out peculiar behaviours in Green functions of the (super) conformal field theories under consideration at short distances. Four-point functions develop logarithmic singularities in the limit of coincident points. This kind of behaviour has already been shown in the instanton contribution to a four-point function in =4 super Yang–Mills theory in section 4.5. Analogous results have N been observed by various authors in Green functions computed from supergravity amplitudes in AdS S5. 5 × Most of the work on the AdS/CFT correspondence has concentrated on the case of AdS S5 which is related to =4 super Yang–Mills. However considering the 5 × N compactification of the type IIB superstring on AdSd+1 M9 d, with M9 d a different × − − compact manifold, one can study superconformal field theories with less supersymmetry. This is a very interesting subject for future developments. In particular the possibility of generalizing the results to non-supersymmetric theories would be of great relevance. Some recent reviews on the subject can be found in [123]. The chapter is organized as follows. In section 5.1 a very brief introduction to type IIB superstring theory and to its low energy supergravity limit is given. The concept of D-branes, that has played a crucial rôle in Maldacena’s construction, is recalled in section 5.2. The original conjecture of [38] together with the successive formalizations are reviewed in section 5.3. The results by various authors in the computation of two- and three-point functions are reported in section 5.4, while section 5.5 contains a discussion on four-point functions. The remaining sections report on original results in the comparison between instanton effects in =4 super Yang–Mills theory and N D-instanton contributions to amplitudes in type IIB superstring theory, following [89].

5.1 Type IIB superstring theory: a bird’s eye view

After their introduction in the late sixties in the context of “dual models” for the description of hadronic processes, string theories have emerged as a prime (and up 5.1 Type IIB superstring theory 143

to now unique) consistent candidate for a uniﬁed theory of fundamental interactions including gravity. A form of the action describing the motion of a string in d-dimensional Minkowski space-time that is suitable for quantization has been proposed in [124, 125]

1 S = dσdτ √γγab∂ Xµ∂ X , (5.1) P 4πα a b µ ′ ZΣ where Xµ = Xµ(σ, τ) are the coordinates of the string which give the embedding of the two-dimensional surface spanned by the string (worldsheet) into the d-dimensional 2 space-time (target space), α′ is a parameter with dimension (mass)− , related to the 1 ab string tension T by T = 2πα′ , and γ is the two-dimensional metric. Roughly speaking the quantization of the bosonic string is achieved by expanding in normal modes the solution of the classical equations of motion which follow from (5.1) and interpreting the coefficients of the expansion as creation and annihilation operators. For open strings this gives rise to a spectrum made of a tachyon with mass m2 = 1 and a massless vector plus an infinite tower of massive states, with masses − α′ proportional to 1 . For closed strings one obtains, beyond the tachyon and a tower √α′ of massive states, a rank two tensor, which decomposes into a symmetric traceless tensor, an antisymmetric tensor and a scalar. The presence of tachyons in the spectrum of bosonic string theories is very problematic. An interesting extension is represented by superstring theories, in which worldsheet supersymmetry is obtained by supplementing the bosonic fields Xµ(σ, τ) with their superpartners, Ψµ(σ, τ), that are two dimensional Majorana spinors and space-time vectors. Superstring theories are consistently formulated in ten space-time dimensions and, as will be briefly discussed, possess a spectrum of states including a finite set of massless states plus an infinite tower of massive states, but no tachyons. The observation that two-dimensional gravity is conformally invariant at the classical level is the key point that allows to exploit the powerful tools of conformal field theory to describe the dynamics of the string. This is achieved by mapping the string worldsheet to the complex plane through (σ, τ) z = eτ+iσ and introducing bosonic → and fermionic coordinates for the string as functions of z and z. The quantization is then performed employing a BRS formalism to identify the physical states in the Hilbert space generated by “vertex operators”. However since the aim of this section is to briefly review the basic ingredients of superstring theories (particularly the type IIB superstring) the more intuitive formalism of oscillators will be used. 144 Chapter 5. AdS/SCFT correspondence

The supersymmetric generalization of (5.1) is

1 µ S = dσdτ √γγab ∂ Xµ∂ X + iΨ ρ ∂ Ψ + 4πα a b µ a b µ ′ ZΣ 1 µ +χ ρbρa X ∂ Xµ + Ψ Ψ χ , a µ b 2 µ b where ρa are worldsheet Dirac matrices. χa is a Rarita–Schwinger ﬁeld, the worldsheet

gravitino (superpartner of γab), which is a space-time scalar. Exploiting worldsheet superconformal symmetry one can put the action in the form

1 2 µ S = d σ [∂+X ∂ Xµ iψL ∂+ψL iψR ∂ ψR] , (5.2) 4πα − − · − · − ′ Z where + and indices refer to light-cone coordinates, σ = τ σ. The equations of − ± ± motion coming from (5.2) lead to an expansion of Xµ and ψµ in terms of independent left- and right-moving oscillation modes for closed strings. For open strings imposing Neumann (free) boundary conditions makes left- and right-moving degrees equivalent, so that roughly speaking the open string has half the degrees of freedom of a closed string.

The coordinate σ is taken in the range [0, 2π] and is periodic; for the ﬁelds ψL,R there are two consistent boundary conditions, either periodic, giving rise to the so called Ramond sector (R), or antiperiodic, corresponding to the Neveu–Schwarz sector (NS). The solution of the equations of motion can be written schematically in the form

µ µ in(τ+σ) ∂XL = α ne (5.3) − n X µ µ in(τ+σ) ψL = ψ ne (5.4) − n X and analogously for the right-moving part. The sum in (5.3) is over integer n, while for the fermions in (5.4) n takes integer or semi-integer values in the R and NS µ sectors respectively. The zero-modes in the expansion of ∂XL,R correspond to the

momentum of the center of mass of the string, PL = PR = P . For the fermions there are only zero-modes in the R sector that satisfy a Cliﬀord algebra. The ground state of the quantum theory is constructed acting with the creation operators associated with the zero-modes on the trivial vacuum. It is a SO(9,1) spinor in the R sector and a (tachyonic) scalar in the NS sector. A generic state in the spectrum of the string is of the form

Lµ1 Lµk Rν1 Rνh α n1 ...ψ nk . . . PL, a α m1 ...ψ mh . . . PR, b , − − | i ⊗ − − | i 5.1 Type IIB superstring theory 145

where conventionally the negative modes are taken as creation operators and a and b refer to the choice of the spin ground state in the R sector. The states are built considering separately the NS and R sectors for left- and right-moving degrees of freedom and then combining them. For the left-movers for instance this construction leads to a tachyonic ground state in the NS sector, while the first higher mass state is a massless vector. In the R sector the ground state corresponds to massless fermions. The correct way of dealing with tachyons was worked out in [36], where a prescription, known as GSO projection, was given which leads to space-time supersymmetric string theories, containing no tachyonic state. In the NS sector the GSO projection eliminates the tachyon and leaves with a massless vector, while in the R sector it produces a spinor with definite chirality. Denoting, after the GSO projection, by v the NS ground state and with s and s′ the spinors with opposite chirality of the R sector one has the following possibilities. If the GSO projections on L and R moving degrees of freedom are different one gets the type IIA superstring theory, which has space-time supersymmetry =(1,1) and is non chiral, whereas using the same N projection on the L and R parts gives the type IIB string theory, which has =(2,0) N supersymmetry and is chiral. In particular for the type IIB theory, on which the attention will be focused in the following, the massless bosonic degrees of freedom are obtained combining

NS NS v v ⊗ ←→ ⊗ or

R R s s , ⊗ ←→ ⊗ and correspond to the following ﬁeld content

NS NS (g , B ,φ) ⊗ ←→ µν µν (0) (4) R R χ = C , B′ ,C , ⊗ ←→ µν where gµν is a rank two symmetric traceless tensor (the graviton), Bµν and Bµν′ antisymmetric tensors, φ a scalar (the dilaton), χ a pseudoscalar (the axion), and C(4) a four-form with self-dual field strength, F (5) = dC(4) = ( F )(5). From the ∗ mixed sectors (R NS and NS R) one obtains the fermionic superpartners of the ⊗ ⊗ previous fields. The low energy effective description of type IIB superstring theory, constructed in terms of the above fields, is type IIB supergravity in ten dimensions, which has chiral 146 Chapter 5. AdS/SCFT correspondence

=(2,0) supersymmetry. There are subtleties in writing down a covariant action for N the model because of the self duality constraint on the four form field strength. The covariant equations of motion of the theory have been derived in [126]. Combining the NS NS scalar and the R R pseudoscalar into a complex field τ, ⊗ ⊗ φ τ = τ1 + iτ2 = χ + ie− , (5.5) and denoting by H and H the field strengths of the NS NS and R R two forms, 1 2 ⊗ ⊗ the low energy action of type IIB supergravity can be written, in the Einstein frame, in the form 1 1 2 S = d10X √g R ∂ τ∂Λτ + F (5) IIB 2κ2 − 2τ 2 Λ − 0 Z 2 1 [τH + H ] [τH + H ]ΛΓΠ + ... , (5.6) −12τ 1 2 ΛΓΠ 1 2 2 2 where κ is the ten-dimensional Newton constant (κ (α′) ). In (5.6) the dots stand 0 0 ∼ for higher derivative terms and the fermionic fields have not been displayed. Here and in the following capital Greek letters refer to flat ten-dimensional space-time. In 2 (5.6) the notation F (5) is schematic, since a covariant kinetic term for the R R ⊗ 4-form cannot be written down naively because of the self-duality constraint. The correct way to deal with this problem was described in [127]. The classical action is invariant under SL(2,R) transformations that act projectively on the complex scalar τ. Under the SL(2,R) group the NS NS and R R antisymmetric tensors form ⊗ ⊗ a doublet. The scalar τ parameterizes the coset space SL(2,R)/U(1)B, where the

U(1)B is an anomalous R-symmetry acting on the fermions. In particular the left 1 chirality gravitino carries charge + 2 under this symmetry and the right-chirality 3 dilatino present in the type IIB spectrum has charge + 2 . At the quantum level the continuous SL(2,R) symmetry is expected to be broken to SL(2,Z) [128]. For the calculations presented in the following the only relevant terms in the eﬀective action are those involving only derivatives of the metric apart from some overall function of the complex scalar. For future purposes it is useful to write these terms in the string frame that is related to the Einstein frame by a Weyl rescaling

(s) φ/2 (E) gµν = e gµν . The leading terms in the derivative expansion that will be considered take the form [114]

4 10 2φ 3 φ/2 4 (α′)− d X√g e− R + κ(α′) f (τ, τ¯)e− . (5.7) 4 R Z 5.2 D-branes 147

where κ is a numerical constant. The Riemann curvature enters the 4 factor in a R manner that may be most compactly described by writing it as an integral over a sixteen-component Grassmann spinor,

4 16 4 d Θ(R 4 ) , R ≡ Θ Z where

Λ1Λ2Λ Λ3Λ4 4 RΘ = ΘΓ Θ ΘΓ ΛΘ RΛ1Λ2Λ3Λ4 , which only includes the Weyl tensor piece of the Riemann tensor. Here, ΓΛ1Λ2Λ3 are the totally antisymmetric products of three ten-dimensional Γ-matrices and the Grassmann parameter Θa (a = 1,..., 16) is a chiral spinor of the ten-dimensional theory. This expresses 4 as an integral over half of the on-shell type IIB superspace. R Beyond these terms there is a sixteen-dilatino term in the eﬀective action that is related to the 4 term by supersymmetry and will play a central rˆole in the R comparison with the results of instanton calculus presented in the previous chapter. The sixteen fermion interaction reads [129, 130]

1 10 φ/2 16 (α′)− d X√g e− f16(τ, τ)Λ + c.c. , (5.8) Z where the complex chiral dilatino has been denoted by Λ and the interaction is (totally) antisymmetric in the sixteen spinor indices. The properties of the modular functions f4(τ, τ) and f16(τ, τ) in equations (5.7) and (5.8) will be discussed later. They receive both perturbative corrections and non-perturbative contributions from D-instantons [114]. The latter will be related to instanton eﬀects in =4 super N Yang–Mills theory.

5.2 D-branes

The perturbative spectrum of type II (A and B) superstring theories does not contain states charged with respect to the ﬁelds in the R R sector [131]. However the ⊗ ﬂurry of work on string dualities during the last years has lead to a much deeper understanding of non-perturbative aspects of string theories, changing dramatically the scenario which was established at the perturbative level. From a low energy viewpoint the existence of solitonic solutions of the supergravity equations of motion has been known for a long time [132]. These solitonic 148 Chapter 5. AdS/SCFT correspondence

configurations describe extended objects, p-branes, that are naturally coupled to the R R forms and their duals. The rôle of these non perturbative states in the fun- ⊗ damental string theory has been clarified by Polchinski [133] through the concept of D-branes. A p-brane is p-dimensional object, whose world-volume is p + 1-dimensional. It naturally couples to a (p + 1)-form, C(p+1), through an action of the form

C(p+1) , ZVp+1

where Vp+1 is the world-volume of the brane. In d space-time dimensions the corresponding “electric” charge would be

(p+1) Qe = dC . d−p−2 ∗ ZS∞ The magnetic dual of a p-brane is a (d p 4)-brane, which couples to a (d p 3)-form − − − − and carries “magnetic” charge

(p+1) Qm = dC . p+2 ZS∞ In particular in the case of the type IIB theory in the R R sector there is a ⊗ scalar, the axion χ, i.e. a zero-form C(0) χ, a rank two anti-symmetric tensor, ≡ (2) (4) (0) Bµν′ , i.e. a two-form C and a four form C . Coupled to C one has a (-1)-brane which is an object localized in space-time, i.e. an instanton in Euclidean space-time. Eﬀects of D-instantons in type IIB superstring theory will be discussed later. The two-form couples to a one-dimensional object, a D-string that should not be confused with the fundamental string which couples to the NS NS two-form. Finally there ⊗ is a 3-brane charged with respect to the four-form C(4). Moreover one can consider the corresponding magnetically dual objects. Starting from the type II supergravity action, extremal (i.e. saturating a BPS bound) solitonic conﬁgurations with the previously discussed properties can be con-

structed assuming that only the metric gµν , the dilaton φ and a p + 1 form are non-vanishing. The ansatz that one considers for these ﬁelds is

ds2 = e2A(r)d~x2 + e2B(r)d~y2 φ = φ(r) C(p+1) = eC(r) , 01...p − 5.2 D-branes 149

where the ten-dimensional coordinates XΛ are split into longitudinal (xµ, µ =0, 1,...,p) i 2 i and transverse (y , i = p +1,... , 9) coordinates and r = yiy . The solution of the supergravity equations can be written in terms of a single harmonic function Hp(r), i.e. a solution of the Laplace equation in the 9 p transverse dimensions, − ap Hp(r)=1+ 7 p , r − 7 p where ap is a constant with dimension L − , related to the charge carried by the p-brane. The p-brane solution is (in the Einstein frame)

p−7 p+1 2 8 2 8 2 ds = [Hp(r)] d~x + [Hp(r)] d~y (5.9) (p+1) 1 C01...p (r) = [Hp(r)]− 1 (5.10) 3−p− 2φ(r) e = [Hp(r)] 2 . (5.11)

Notice in particular that in the case of a 3-brane the dilaton φ reduces to a constant. The solution simpliﬁes in the string frame where the metric becomes

1 1 2 2 2 2 2 ds = [Hp(r)]− d~x + [Hp(r)] d~y .

It can be proved that these solitonic solutions are BPS saturated and preserve one half of the supersymmetries of the original flat background, i.e. 16 supercharges out of 32. This implies for instance that there exist a relation between the mass and charge of the solitons. Many of the recent developments in string theory have been made possible by the suggestion [133] of a way to incorporate the above solitonic objects into type II string theories, see [134] for a review. The solitonic p-branes are introduced in the microscopic string theory in relation with a (Dirichlet) open string subsector. More precisely one introduces Dirichlet p-branes (or Dp-branes) as p-dimensional hyperplanes where open strings can end, satisfying Neumann (standard) boundary conditions on the p longitudinal coordinates Xµ, µ = 0,... ,p 1 and Dirichlet (fixed) conditions on the remaining transverse − coordinates Xm, m = p,... , 9. D-branes defined in this way can be seen as ‘topological defects’ where open strings can end. These hyperplanes are actually dynamical objects, whose elementary excitations are described by the open strings attached to them. These open strings can be consistently introduced in type II superstring theories, without considering ‘standard’ open strings propagating independently in the target space. There are various arguments that lead to identify D-branes with 150 Chapter 5. AdS/SCFT correspondence

the solitonic p-branes of type II supergravities. For instance considering two parallel D-branes their interaction is mediated by the exchange of closed strings and one can prove that this results in a vanishing total force just like for example is found for a monopole pair. Furthermore it can be shown that the boundary conditions imposed on the open strings ending on the D-brane break half the supersymmetries, so that the D-branes are BPS saturated objects. The dynamics of D-branes, in a microscopic string theory perspective, is described by the open strings attached to the brane. Comparing the effective world-volume dynamics of the D-brane with the dynamics of solitonic p-brane solutions in the moduli space approximation one can find further evidence for the identification of the two. The massless spectrum of the subsector of open strings attached to a p D-brane is a maximally supersymmetric U(1) gauge theory in p + 1 dimensions. The effective world-volume degrees of freedom of a p D-brane correspond to the dimensional reduction of a =1 vector multiplet from d=10 to d = p + 1. This gives rise to a N supersymmetric Maxwell theory (with sixteen supercharges) including 9 p scalars, − one vector and their superpartners. Considering N D-branes one can have also open strings stretched between them. They describe the exchange of massive vector states. There are N 2 ways of stretching the strings among the branes. In the limit in which the N D-branes are taken to coincide these N 2 different states become massless. The symmetry is enhanced to U(N) and one is left with an effective U(N) Yang–Mills theory in which the N 2 vectors form the adjoint. In general the relative separations among the branes determine the expectation values of the 9 p scalars in the effective theory. − In particular in the case of a collection of N coincident D3-branes the world- volume effective action is a U(N) supersymmetric Yang–Mills theory with sixteen supercharges in d=4, i.e. the =4 super Yang–Mills theory with gauge group U(N) 1. N

1Actually there is a U(1) factor that describes the center of mass motion and decouples, so that the gauge group is SU(N) [135]. 5.3 AdS/SCFT correspondence 151

5.3 The AdS/SCFT correspondence conjecture

In the case of D3-branes the solution for the metric yields (in the string frame)

1 1 L4 − 2 L4 2 ds2 = 1+ d~x2 + 1+ d~y2 , (5.12) r4 r4 where (xµ,yi) are the previously introduced Cartesian coordinates and L is a length scale. As already noticed in the case of a D3-brane the dilaton is constant, φ = φ0, and determines the string coupling constant

φ0 gs = e .

Considering a D3-brane configuration with N units of R R 5-form field strength, ⊗ i.e. a configuration of N coincident D3-branes, the length L is

4 2 L =4πgsNα′ . (5.13)

The key observation which led Maldacena to the formulation of the AdS/CFT correspondence conjecture [38] is that the metric (5.12), in the “near horizon” limit r 0, → reduces to L2 ds2 = dx dx + dρ2 + dω2 , (5.14) ρ2 · 5 2 L4 2 where ρ = r2 and dω5 is the spherically symmetric constant curvature metric on a 5-sphere, which is the metric of the AdS S5 space. In (5.14) L is the radius 5 × of curvature of both the 5-sphere and the AdS factor. The D3-brane metric can be viewed as a solitonic configuration interpolating between two maximally supersymmetric backgrounds, the near horizon AdS S5 and flat ten dimensional Minkowski 5 × space-time at infinity. Maldacena has argued, on the basis of previous results in the context of the description of black holes by D-branes [136], that the region that should be identified with the Yang–Mills low energy description is the “throat”, i.e. the region r ≪ L, in the D3-brane metric (5.12). See [137] for related work. This leads to the conjecture that =4 supersymmetric Yang–Mills theory should be dual to type IIB N superstring theory compactified on AdS S5. The original conjecture relates type 5 × IIB supergravity to SU(N) =4 super Yang–Mills in the large N limit. This is N related to the fact that the limit g 1 at large length scale with respect to the s ≪ 152 Chapter 5. AdS/SCFT correspondence

′ string scale, α 1, in which one can trust the supergravity approximation, requires L2 ≪ N with 4πg N fixed and large. The correspondence relates the complexified → ∞ s Yang–Mills coupling constant to the one of the type IIB superstring in AdS S5 by 5 × g2 θ g = YM , χ = YM , (5.15) s 4π 0 2π where χ is the constant R R scalar and from now on the Yang–Mills coupling 0 ⊗ constant and vacuum-angle will be denoted respectively by gYM and θYM . This implies that the large N limit of relevance here coincides with the large N 2 limit considered by ’t Hooft in [122], with the effective couplingg ˆ = gYM N fixed at a large value.

The AdS5 space has constant negative curvature and possesses a boundary that is the four dimensional Minkowski space 2. According to Maldacena’s proposal, the boundary of AdS , which corresponds to ρ 0, i.e. r in the notation of (5.14), 5 → →∞ is exactly the location of the =4 theory and the boundary values of the bulk N supergravity fields act as sources that couple to gauge-invariant composite operators in =4 super Yang–Mills theory. The correct interpretation of this statement has N been explained in [118, 119] and will be discussed soon. The AdS S5 background is characterized by the non-vanishing fields, 5 × 1 1 F = ε R = (g g g g ) MNPQR L MNPQR MNPQ −L2 MP NQ − MQ NP 1 1 F = ε R =+ (g g g g ) , mnpqr L mnpqr mnpq L2 mp nq − mq np where upper case Latin indices, M,N, = 0, 1, 2, 3, 5, span the AdS coordinates ··· 5 and lower case Latin indices, m, n, = 1, 2, 3, 4, 5 span the S5 coordinates. The ··· only non-vanishing components of the Ricci tensor are 4 4 R = g R =+ g . (5.16) MN −L2 MN mn L2 mn Upon contracting (5.16) with the metric tensor it follows that the total (5+5)- dimensional scalar curvature vanishes. The Weyl tensor, defined in d-dimensional space-time as 1 C = R [R g + (3 terms)] + µνρσ µνρσ − d 2 µν νσ 1 − + R (g g g g ) , (d 1)(d 2) µρ νσ − µσ νρ 2 − − Actually this is better understood considering the Euclidean version of the AdS5 space, see [119] for a detailed description. 5.3 AdS/SCFT correspondence 153

vanishes as well in AdS S5 because of the conformal ﬂatness of the metric. 5 × This background is maximally supersymmetric (just like the Minkowski vacuum) so there are 32 conserved supercharges, that transform as a complex chiral spinor of the tangent-space group, SO(4,1) SO(5). In the basis where the ten dimensional Γ × Λ matrices are given by Γ = σ γ 1Iand Γ = σ 1I γ , the supersymmetries M 1 ⊗ M ⊗ m 2 ⊗ ⊗ m are generated by the Killing spinors that satisfy 1 D ǫ (σ 1I 1I)Γ ǫ =0 , (5.17) Λ − 2L 1 ⊗ ⊗ Λ which follows from the requirement that the gravitino supersymmetry transformation should vanish. In this basis the complex chiral supersymmetry parameters read

1 ǫ = ζ κ , ± 0 ⊗ ± ⊗ ± where ζ are complex four-component SO(4,1) spinors and κ complex four-component ± ± SO(5) spinors, satisfying 1 DM ζ γM ζ = 0 ± ∓ 2L ± 1 Dmκ i γmκ = 0 . ± ∓ 2L ±

Strong support to the conjecture comes from symmetry considerations. AdS5 has a SO(2,4) isometry group which is to be identified with the four-dimensional conformal group in the boundary theory. The isometry group of the S5 factor is SO(6) SU(4) and is related to the SU(4) R-symmetry group of the =4 Yang–Mills ∼ N theory. Including the associated fermionic generators the super-isometry group of the type IIB background (5.14) is exactly the superconformal group SU(2,2 4) of =4 | N super Yang–Mills in four dimensions. Moreover the SL(2,Z) S-duality group that is conjectured to be preserved in the type IIB superstring at the quantum level is identified with the (conjectured) exact S-duality symmetry group of four dimensional SU(N) =4 Yang–Mills theory. The former is connected to the existence of an N infinite set of stable dyonic BPS string solitons [128], the latter to the existence an infinite set of stable dyonic BPS states [72], as discussed in chapter 2. The spectrum of excitations of the maximally supersymmetric compactification of the type IIB superstring on the AdS S5 background, first considered in [126], was 5 × studied in great detail in [138, 139]. The map that associates fields in the multiplet of type IIB supergravity to the corresponding operators in =4 super Yang–Mills N 154 Chapter 5. AdS/SCFT correspondence

has been worked out in [140, 141, 142]. States in the spectrum of d=5 =8 gauged N supergravity admit a natural classification in terms of spherical harmonics on the 5-sphere. In the conjectured correspondence the lowest lying states, assembled into the =8 gauged supergravity multiplet, couple to the supermultiplet of super Yang– N Mills currents, equations (2.17) and (2.18), that are “bilinear” in the elementary Yang–Mills fields. Higher ℓ spherical harmonics, i.e. Kaluza–Klein (KK) descendants of the supergravity fields, couple to other “chiral primary fields”, that correspond to “ℓth-order” gauge-invariant polynomials in the elementary super Yang–Mills fields. Moreover, stable string excitations that couple to non-chiral primary fields are needed for the closure of the operator algebra [143]. Despite some interesting progress [144], a useful worldsheet description for the type IIB superstring in AdS S5 is still 5 × lacking. Nevertheless the stringy truncation of the KK spectrum to ℓ N seems to ≤ be a robust result that matches with the naive expectations on the super Yang–Mills side. The explicit connection between the bulk theory and the boundary theory has been given a precise formulation in [118, 119]. The partition function of the type IIB superstring, computed with suitable boundary conditions on the four-dimensional

boundary of AdS5, plays the rˆole of generating functional for connected Green functions of gauge-invariant composite operators in =4 Yang–Mills theory. More pre- N cisely one identiﬁes

Z [J]= [DA] exp( S [A]+ [A]J) , (5.18) IIB − YM O Z

where ZIIB [J] is the partition function of the type IIB superstring, evaluated in terms of the bulk string fields, which is computed with prescribed boundary values J for the fields. Elementary fields of the =4 super Yang–Mills theory on the boundary N are denoted by A and (A) are gauge-invariant composite operators to which J(x)’s O couple. The left hand side of (5.18) is computed performing the functional integration over the bulk fields Φ(z; ω), where ω are the coordinates on S5 and zM (xµ, ρ) ≡ (M =0, 1, 2, 3, 5 and µ =0, 1, 2, 3) the coordinates on AdS (ρ z is the coordinate 5 ≡ 5 transverse to the boundary)

Z [J]= [DΦ] exp( S [Φ]) . (5.19) IIB J − IIB Z In (5.19) the notation Φ(z; ω) refers collectively to the ‘massless’ supergravity ﬁelds, 5.3 AdS/SCFT correspondence 155

their Kaluza–Klein descendents, and possibly the string excitations. The functional integral depends on the boundary values, J(x), of the bulk fields. The exact map which associates fields in the type IIB compactification with operators in =4 super Yang–Mills theory is dictated by the matching of quantum N numbers corresponding to the previously discussed global symmetries. Furthermore the conformal dimension, ∆, of a Yang–Mills operator is related to the AdS ‘mass’ O m of the corresponding supergravity field. For instance in the case of scalar operators the relation is

(mL)2 = ∆(∆ 4) , (5.20) − where only the positive branch of ∆ = 2 4+(mL)2 is relevant for the lowest- ± ± ‘mass’ supergravity multiplet. p As already remarked in the Kaluza–Klein reduction of type IIB supergravity on S5 the states are classified expanding the ten-dimensional fields in spherical harmonics on the 5-sphere. This procedure generates an infinite tower of states with spins up to 2. The masses of the states associated with single modes and their behaviour under the SO(2,4) isometry group of AdS5 are determined by solving the corresponding linearized equations of motion [139]. Alternatively the Kaluza–Klein excitations of all the fields in the type IIB gauged supergravity can be constructed, using the Killing spinors defined in equation (5.17), from the modes of the massless singlet dilaton eφ. The construction is rather involved and will be briefly sketched only for fields that will be relevant in the following. One obtains a classification of the states according to the representations of the isometry group of AdS S5, which adding the associated fermionic generators be- 5 × comes SU(2,2 4). In particular the mass of the states is identified with the eigenvalue | of one of the generators of SO(2,4). This allows to construct a map which relates each field in the supergravity multiplet to the corresponding composite operator in =4 N super Yang–Mills theory by matching the SU(2,2 4) quantum numbers (recall that | on the super Yang–Mills side SU(2,2 4) is the =4 superconformal group). Notice | N that since the states in the supergravity multiplet as well as their KK descendants have spins not larger than 2, they belong to short multiplets of SU(2,2 4). | The massless dilaton is associated with the constant mode on S5, i.e. with the scalar spherical harmonic Yℓ(ω) with ℓ = 0. Also at the ℓ = 0 level there is the

ﬁve-dimensional graviton, describing the excitations of the AdS5 metric. The other scalars in the supermultiplet are associated with excitations on the 5-sphere, i.e. they 156 Chapter 5. AdS/SCFT correspondence

ij 2 4 have ℓ > 0. In particular there are real scalars Q with mass m = in the 20R − L2 of the SO(6) isometry group of S5. They result from a combination of the trace of the internal metric and the self-dual R R five-form field, F (5), with ℓ =2(Qij are ⊗ quadrupole moments of S5). The trace part of the metric fluctuations can be written in the form 16 δg = fδ δg = fδ , MN 15 MN mn mn where f = f(z,ω). Analogously for the R R 4-form one gets ⊗ L L A = ε Rf A = ε rf . MNPQ − 4 MNPQR∇ mnpq − 4 mnpqr∇

The relation with the scalar ﬁelds Qij(z) is of the form

ij f(z,ω)= Qij(z)Yℓ=2(ω) .

Similarly the complex scalars EAB with mass m2 = 3 and their conjugates are − L2 associated with the pure two-form fluctuations with ℓ = 1 of the complexified antisymmetric tensor, constructed from the R R and NS NS two-forms, in the [ij⊗] ⊗ internal directions. The massless vectors VM in the 15, that gauge the SO(6) isometry group, are in one-to-one correspondence with the Killing vectors of S5 and result from a linear combination with ℓ = 1 of the mixed components of the metric and the internal three-form components of the R R four-form potential, C(4). The 6 ⊗ [AB] 2 1 complex antisymmetric tensors BMN with m = L2 , that have peculiar first order equations of motion, result from scalar spherical harmonics with ℓ = 1. The analysis of the fermions is similar. The dilatini ΛA in the 4 of SO(6) are proportional to the internal Killing spinors κ and have mass m = 3 . They play a central rôle in the + − 2L comparison between the effects of instantons in =4 Yang–Mills and D-instantons in N type IIB string theory that has been carried out in [89] and will be reviewed in section A 1 5.6. The 20C spinors χ with mass m = correspond to internal components BC − 2L of the gravitino with ℓ = 1. Finally the supergravity multiplet is completed by the

massless 4 gravitinos ΨMA which are proportional to the internal Killing spinors κ . − The above ﬁelds are the ones that act as sources for the superconformal currents (2.17), (2.18). More precisely the graviton and dilaton are associated with the SU(4) singlets in the =4 current multiplet, the stress-energy tensor and the N Tµν scalar respectively. The other scalars, EAB and Qij, correspond to AB and ij C E Q in (2.18) respectively. The massless vectors V [ij] are dual to the SU(4) currents A M JµB 5.3 AdS/SCFT correspondence 157

and the anti-symmetric tensors B[AB] to AB. The identiﬁcation can be carried out MN Bµν A A analogously for the fermions. The dilatini Λ and the spinors χBC are associated ˆ A A respectively to the fermions Λ andχ ˆBC in (2.18). The gravitinos ΨMA correspond to the supersymmetry currents ΣµA. Higher Kaluza–Klein modes correspond to higher values of ℓ. For example, there are other scalar modes with ℓ> 2. Each of these can be put in one-to-one correspondence with a gauge singlet composite operator W(ℓ) that starts with

W (i1...iℓ) = tr φ(i1 ...φiℓ) traces , (5.21) θ=0 − which has dimension ∆ = ℓ and belongs to the ℓ-fold symmetric traceless tensor representation of SO(6) with Dynkin labels (0, ℓ, 0). The multiplet contains 256ℓ2(ℓ2 − 1)/12 states with different ∆ and SU(4) quantum numbers. The correct recipe for calculating correlations follows from (5.18) and amounts to computing the truncated Green functions in the bulk and attaching bulk-to-boundary propagators to the external legs [118, 119, 145]. The precise forms of the latter depend on the spin and ‘mass’ of the field. For instance, the bulk-to-boundary Green function for a bulk scalar field Φ associated to a dimension ∆ operator on the boundary m O∆ reads ∆ µ µ ρ K (x , ρ; x′ , 0) = c , (5.22) ∆ ∆ (ρ2 +(x x )2)∆ − ′ where c is a constant given by c = Γ(∆)/[π2Γ(∆ 2)]. For simplicity the ω ∆ ∆ − dependence has been suppressed, so that the expression (5.22) is appropriate for an ‘S-wave’ process in which there are no excitations in the directions of the five-sphere, 5 S . In terms of K∆ the bulk field

4 Φm(z; J)= d x′K∆(x, ρ; x′, 0)J∆(x′) (5.23) Z 4 ∆ ∆ 4 satisfies the boundary condition Φ (x, ρ; J) ρ − J (x) as ρ 0, since ρ − K m ≈ ∆ → ∆ reduces to a δ-function on the boundary. As observed in [145] the bulk-to-boundary propagator must be modified for ∆=2. In this case the previous expression vanishes since c =0 for ∆=2. In the special case ∆=2 one must require Φ (x, ρ; J) ∆ m → 2 Γ(2) ρ log ρJ(x), which yields c∆=2 = 2π2 . Before discussing specific calculations, that will be related to instanton effects in =4 supersymmetric Yang–Mills theory, this N prescription for evaluating Green functions will be illustrated in more detail in the case of two- and three-point functions in the next section. 158 Chapter 5. AdS/SCFT correspondence

The generalization of this construction to fermionic fields has been considered in [146] and will be relevant later when the calculation of a correlation function of sixteen fermionic dilatini will be presented. The original conjecture involves the supergravity approximation and holds in the large N limit. However in [38] it has been suggested that the correspondence can be generalized to finite N by taking in account string theory corrections. This claim has been made more explicit in [119]. For finite N the rôle of generating functional is played by the full partition function of the string in AdS S5, computed with 5 × suitable boundary conditions. Moreover for finite N, i.e. for finite radius, one expects a truncation to ℓ N of the Kaluza–Klein spectrum, that should result by taking ≤ into account full-fledged stringy geometry as in any string compactifications. Perturbative checks of the conjecture at finite N for two- and three-point functions have been given in [147, 148] and will be discussed in the next section. The non- perturbative computations of [89] suggest that the correspondence remain valid at finite N also for four- or higher-point functions of some protected operators at least. Witten has also suggested an intuitive way to represent diagrammatically the above described construction through graphs like those depicted in figure 5.1.

Figure 5.1: Witten’s representation of three point functions in the AdS/CFT correspondence.

The circles in figure 5.1 represents the four-dimensional boundary of AdS5 and the interior is the bulk. Lines connecting an internal point with a point on the boundary are the above discussed bulk-to-boundary propagators (see e.g. equation (5.22)). In 2 the supergravity approximation, i.e. in the limit of large N and large and fixed gYM N, only tree diagrams like the first one in figure 5.1 contribute, while the loop-diagrams become relevant for finite N, when string effects must be taken into account. This 5.3 AdS/SCFT correspondence 159

can be understood as the loops correspond to closed string exchange which gives a contribution that is suppressed in the large N limit. Attempts have also been made to generalize the correspondence to conformal theories with less supersymmetry and hopefully to non-supersymmetric theories. This is achieved by studying the compactification of the type IIB superstring on AdS X , 5 × 5 where X5 is a compact Einstein manifold, i.e. a positively curved compact manifold with metric satisfying Rµν = λgµν , with λ> 0. The simplest examples are obtained 5 considering X5=S /Γ, where Γ is a discrete subgroup of SO(6). In this case X5 presents orbifold type singularities, but it has locally the geometry of S5. In [149, 150] this models have been discussed in detail and it has been shown that taking N coincident D3-branes placed at an orbifold singularity one can construct =2 and N =1 theories. Such theories have product gauge groups SU(N)k and contain matter N fields in the fundamental and adjoint representations. A possible approach to the study of non conformal field theories through the AdS correspondence has been suggested in [151]. The model proposed in [151] is based on the observation that field theories at finite temperature can be described by near-extremal p-brane solutions. Non supersymmetric theories are then obtained by imposing anti-periodic boundary conditions for the fermions on the compactified direction, corresponding to the raising of temperature, which explicitly break supersymmetry. Qualitative agreement with the expectations for QCD at strong coupling has been found following this approach. In particular the results from supergravity calculations include the area low behaviour of Wilson loops, the presence of a mass gap for glueball states, the relation between confinement and dual superconductivity and the existence of heavy quark baryonic states [151, 152]. Numerical calculations of glueball masses have been performed in [153] applying the same method. The results appear to be in agreement with those obtained from lattice simulations. The limit of this approach is that for the supergravity approximation to be reliable one must take the effective ’t Hooft coupling to be large, while the weak coupling region is the one relevant for the continuum limit on the lattice. As a consequence to actually compare the results from AdS with lattice extrapolations to the continuum, it seems to be necessary to extend the correspondence to finite N, which requires a full string theory description in AdS. More recently the near-horizon geometry of a stack of D3-branes in type 0 theories has been proposed as a candidate for a non supersymmetric version of the above discussed correspondence. In these models the “tachyon should . . . peacefully 160 Chapter 5. AdS/SCFT correspondence

condense in the bulk” [154]. The basic idea suggested by Polyakov in [154] is that non-supersymmetric theories should have as their duals string theories displaying worldsheet supersymmetry, but no space-time supersymmetry. This is the property characterizing type 0A and 0B string theories, which have been introduced in [155]. This theories are constructed by a non-chiral GSO projection, which explicitly breaks space-time supersymmetry and leads to a spectrum containing no space-time fermions. The bosonic spectrum is the same as in the corresponding type II theories in the NS NS sector and contains a doubled set of R R fields. The interest in these ⊗ ⊗ theories has been renewed in the systematic construction of their open string descendants [156]. The inclusion of D-branes in such theories has been studied in [156, 157] and allows to construct non-supersymmetric gauge theories as world-volume theories, which do not contain open string tachyons [156, 154, 158]. In particular in [158] a SU(N) Yang–Mills theory coupled to six real massless scalars has been constructed as world-volume theory of N coincident D3-branes in type 0B theory. In this model the 5-form field strength is not constrained to be self dual and so one can consider both electrically and magnetically charged D3-branes. Moreover in the D3-brane solution the dilaton is not constant, but acquires a spatial (radial) dependence as a consequence of the presence of the NS NS tachyon. In [158] it is argued that this radial ⊗ dependence may be related to a dependence of the coupling constant on the energy scale in the gauge theory description. In [159, 160] a field theory with asymptotically free behaviour is constructed from a D3-brane type 0 background with AdS S5 5 × geometry. This very interesting issues will not be further addressed here, in the following the attention will be focused on the correspondence between type IIB string theory in AdS S5 and =4 super Yang–Mills theory. 5 × N

5.4 Calculations of two- and three-point functions

In this section the prescription for the calculation of correlation functions will be discussed in more detail. First the general idea in the simplest case of scalar ﬁelds and for spinors will be reviewed considering two-point functions, then some general considerations on the properties of correlators of R-current operators and some non- renormalization theorems will be reported. Finally the computation of two- and three-point functions of generic chiral primary operators will be discussed. The computation of two-point functions according to the prescription of the pre- 5.4 Two- and three-point functions 161

vious section goes as follows. The Euclidean action for a real massive scalar ﬁeld in

AdS5 is 1 S = d5z √g gMN (∂ Φ)(∂ Φ) + m2Φ2 . (5.24) 2 M N Z As has been explained in [118, 119] in order to compute the two-point Green function of an operator [A], related by the correspondence to Φ and coupled to a source J in O the boundary theory, in the supergravity approximation valid for large N, one must solve the equation of motion obtained from (5.24) with the condition that Φ(x, ρ) approach J(x) as ρ 0. The equation of motion is →

1 MN 2 ∂M √gg ∂N Φ m Φ=0 . √g − Then, after substituting into (5.24) the solution with the prescribed behaviour in the limit ρ 0, taking the functional derivatives with respect to J one obtains the Green → function. The relevant solution is of the form of equation (5.23). As a result one gets for the two-point functions

4 d x′dρ 2 M (x) (y) = ∂ K (x′, ρ; x, 0)ρ ∂ K (x′, ρ; y, 0)+ hO O i − ρ5 M ∆ ∆ Z 2 +mK∆(x′, ρ; x, 0)K∆(x′, ρ; y, 0) . (5.25) Then integration by parts and use of the equation of motion yields 3

Γ(∆ + 1) 1 (x) (y) = . (5.26) hO O i π2Γ(∆ 2) (x y)2∆ − − This computation of two point functions has led Witten to establish the relation (5.20) between the AdS ‘mass’ m of the field Φ and the conformal dimension ∆ of the corresponding operator . In [119] the same analysis has been carried out for massive O vector fields. In general for a p-form field the relation between the AdS ‘mass’ and the scaling dimension of the dual composite operator becomes (∆ 4+p)(∆ p)=(mL)2, − − which yields

∆=2 4+(mL)2 + p(p 4) . ± − p The case of the graviton gMN , related to the stress-energy tensor, has been considered in [118, 161]. 3Again this expression must be modiﬁed in the case ∆=2, see the previous section. 162 Chapter 5. AdS/SCFT correspondence

Repeating the above construction for a massive spinor in AdS5 provides the bulk to boundary propagator for fermionic ﬁelds [146]. One starts from the action for

‘massive’ spinors in AdS5

S = d5z √g Ψ(D/ m)Ψ , (5.27) − Z 1 which yields the Dirac operator acting on spin- 2 ﬁelds in AdS5

M Lˆ 1 Mˆ Nˆ 5ˆ µˆ 5ˆ D/ Ψ= eˆ γ ∂ + ω γ ˆ ˆ Ψ=(ργ ∂ + ργ ∂ 2γ )Ψ , (5.28) L M 4 M MN 5 µ −

M Mˆ Nˆ where eLˆ is the vielbein, ωM the spin connection (hatted indices refer to the tangent space) and γµˆ are the four-dimensional Dirac matrices. Just like in the case of scalar fields in order to compute correlation functions one must solve the equations of motion with assigned boundary conditions. However the action (5.27) vanishes for any field configuration which satisfies the equations of motion. This implies that the partition function is actually independent of the boundary condition on the field Ψ, so that it cannot be used as a generating functional. To solve this problem a suitable modification of the action (5.27) by the addition of a boundary term has been proposed in [146]. The uniqueness of such a boundary term has been recently discussed in [162]. The suggestion of [146] is to supplement the action (5.27) by a term of the form

4 ǫ Sb = lim d x√g ΨΨ , (5.29) ǫ 0 ǫ → ZM ǫ where M is a closed four-dimensional submanifold of AdS5, which approaches the boundary of AdS as ǫ 0. In (5.29) gǫ is the metric on M ǫ induced by the one 5 → of AdS5. This boundary term does not modify the equations of motion in the bulk

and furthermore is invariant under the isometry group of AdS5. As a consequence of the vanishing of the bulk action on field configurations satisfying the equations of motion the entire contribution to the partition function, to be employed as generating functional in the AdS/CFT correspondence, comes from the boundary term (5.29). F This allows to construct the bulk to boundary propagator K∆(x′, ρ; y, ρ0) for spinors of mass m. For fermionic fields, as discussed in [146], the relation between the AdS ‘mass’ of a supergravity field and the scaling dimension, ∆, becomes

∆=2 mL . − 5.4 Two- and three-point functions 163

3 The case of the dilatino, Λ, with AdS mass m = 2L , corresponding to an operator 7 − of dimension ∆ = 2 will be studied in section 5.6. In the construction of [146] there is a subtlety related to the apparent arbitrariness of the coefficient of the boundary term (5.29) to be added to to the bulk action. This point has been recently clarified in [162]. In the derivation of the equations of motion from a variational principle only the negative chirality part of the boundary limit of the spinor Ψ is fixed. The variation of the positive chirality component of Ψ on the boundary generates the boundary term in the action, which is uniquely fixed by the requirement that the total variation of the action must be zero on the solution of the equations of motion in the class of histories defined by the boundary conditions. To compute three-point functions one must consider interaction terms in the supergravity action and solve the corresponding equations of motion, with a suitable prescription for the behaviour on the boundary. For instance, in the case of scalar fields, interaction terms of the form

(A) = Φ Φ Φ L 1 2 3 (B) = gMN Φ (∂ Φ )(∂ Φ ) L 1 M 2 N 3 give the following contributions to three point functions 4 (A) d x′dρ (x) (y) (z) = K (x′, ρ; x, 0)K (x′, ρ; y, 0) hO1 O2 O3 i − ρ5 ∆1 ∆2 · Z K∆3 (x′, ρ; z, 0) 4 · (B) d x′dρ (x) (y) (z) = K (x′, ρ; x, 0)∂ K (x′, ρ; y, 0) hO1 O2 O3 i − ρ5 ∆1 M ∆2 · Z 2 M ρ ∂ K (x′, ρ; z, 0) , (5.30) · ∆3 where ∆1, ∆2 and ∆3 are the dimensions of the operators coupled to Φ1, Φ2 and

Φ3 respectively. Equations (5.30) are obtained, analogously to the expressions for two-point functions, by solving the equations of motion in the presence of the above interaction terms and then substituting back into the action, whose exponential is used as generating functional. To derive (5.30) one must solve the equations recur- sively as a series in the coupling constant gs. To generate three-point functions it is suﬃcient to stop to the lowest order, so that the bulk to boundary propagators, K∆i in (5.30) are the same entering two-point functions. The contributions to three-point functions can be represented through a diagram like the one depicted in ﬁgure 5.2, where the derivatives, read from the interaction, eventually act on the bulk to boundary propagators. 164 Chapter 5. AdS/SCFT correspondence

Figure 5.2: AdS diagram for the correlator of three scalar operators

By writing explicitly (5.30) one immediately sees that these correlation functions have the correct form required by conformal symmetry. Actually, as will be discussed in the case of R-currents, two-point functions in the boundary conformal ﬁeld theory are determined by superconformal symmetry up to a constant, so that the comparison

with the results obtained from AdS5 only allows to establish the map between the ﬁelds on the two sides of the correspondence and to ﬁx the normalizations. Analo- gously three-point functions are strongly constrained. Their supergravity calculation

in AdS5 and comparison with field theory only provides a check of the normalization constants (OPE coefficients). A completely non-trivial check of the proposed correspondence requires the computation of four- or higher-point functions, that will be examined in subsequent sections. The R-symmetry conserved currents of =4 super Yang–Mills theory were given N in equation (2.17). The two-point function a b is completely fixed by supercon- hJµ Jν i formal invariance. The dimension of the operators a is ∆=3 and the form of the Jµ correlator of two currents is [145]

δab 1 a(x) b(y) = c (δ 2 ∂ ∂ ) , hJµ Jν i (2π)4 µν − µ ν (x y)4 − where c is a positive constant, which is only determined by one-loop corrections thanks to a non-renormalization theorem. c is the central charge of the OPE JJ and equals the trace anomaly. Since the latter is related by =1 supersymmetry to N the R-current anomaly which is one-loop exact c is determined by the one-loop result as well. In the case of =4 super Yang–Mills with gauge group SU(N) one ﬁnds N [145] 1 c = (N 2 1) . 2 − 5.4 Two- and three-point functions 165

The three point function of currents is also strongly constrained by superconformal symmetry; it contains a normal parity part and an abnormal parity part [145]

a b c a b c a b c µ (x) ν (y) λ (z) = µ (x) ν (y) λ (z) + + µ (x) ν (y) λ (z) = hJ J J i hJ J J i hJ J J i− abc (+) (+) ( ) abc = f k1 Dµνλ(x, y, z)+ k2 Cµνλ(x, y, z) + k − d Mµνλ(x, y, z) ,

h i abc abc where Dµνλ, Cµνλ and Mµνλ are known tensor functions and the f and d SU(N) a b c 1 abc abc (+) (+) ( ) symbols are deﬁned by tr T T T = 4 (if +d ). k1 , k2 and k − are numerical (+) constants; k1 is ﬁxed by the Ward identity connecting two- and three-point func- (+) c (+) tions, which relates it to c by k1 = 16π2 , k2 is independent and it is not protected ( ) by general non-renormalization theorems. The third constant, k − , is controlled by the Adler–Badeen theorem which implies that it is one-loop exact. The same non- renormalization properties of the R-currents should of course hold true for the other composite operators in the current multiplet. The computation of two- and three-point functions of chiral primary operators (CPO’s) will now be reviewed following [148, 145]. The chiral primary operators that will be considered are of the form

I = gℓ tI tr ϕi1 (x) ...ϕiℓ (x) , (5.31) Oℓ YM i1...iℓ where ϕi are the six real matrix-valued scalar fields of =4 supersymmetric Yang– N Mills theory and tI is a totally symmetric (traceless) rank ℓ tensor of SU(4) SO(6), i1...iℓ ∼ I1 I2 i1...iℓ I1I2 whose normalization is chosen such that ti1...iℓ t = δ . The free propagator for the fields ϕi is δijδ 1 ϕi ϕj = ab . h a bi (2π)2 (x y)2 − Chiral primary operators (5.31) belong to representations of the SU(4) R-symmetry group characterized by Dynkin labels (0, ℓ, 0), they are Lorentz scalars, i.e. they belong to the representation j =j =0 of the Euclidean Lorentz group SO(4)=SU(2) 1 2 × SU(2), and their conformal dimension is ∆=ℓ. The class of CPO’s comprises the operators ij and in particular and considered in section 4.5. Q Q Q In general operators involved in the AdS/CFT correspondence have a conformal dimension ∆ that is not an integer. Only operators belonging to short supermultiplets of the SU(2,2 4) superconformal group, which are dual to the massless AdS | supergravity fields and their KK descendants, have a dimension that is protected under renormalization [163]. In supersymmetric field theories there are chiral primary 166 Chapter 5. AdS/SCFT correspondence

operators, like hose considered here, whose dimensions are integer and protected, even if they are not conserved currents. In the case of =4 super Yang–Mills theory N under consideration the conformal dimension of a generic operator satisﬁes the bound

∆ q = ℓ , (5.32) ≥ where q is the charge of the field under a U(1) subgroup of the SU(4) R-symmetry group. (5.32) is a “BPS-type” bound, which follows from the superconformal algebra (2.16). As already discussed in the previous section the supergravity fields and the Kaluza– Klein states in d=5 =8 gauged supergravity belong to short multiplets of the super- N isometry group SU(2,2 4). The same is true for their counterparts in =4 Super | N Yang–Mills theory (where SU(2,2 4) is the superconformal group). The CPO’s intro- | duced above play a central rôle in the construction of these multiplets. As explained in [163] all these multiplets can be obtained starting from scalar chiral primary operators like those in (5.31) and resorting to analytic superspace. The generic short multiplet is described by a composite twisted chiral superfield [111] of the form

= tr W (i1 ...W iℓ) traces , W(ℓ) −

i 1 i [AB] where the superﬁelds W = 2 tABW are the same introduced in chapter 2, equations (2.7) and (2.8). Massive string states on the contrary are in long multiplets as well as their duals, which are non chiral primary operators. An example of operators of this kind is represented by the =4 embedding of the Konishi multiplet [164, 163], which, in N terms of =1 superﬁelds, is N I V Σ = Φ e ΦI† , (5.33)

where ΦI , I =1, 2, 3 are the =1 chiral multiplets and V the =1 vector multiplet N N of =4 super Yang–Mills theory. N In [148] two- and three-point Green functions of CPO’s have been computed using

type IIB supergravity in AdS5 and Witten’s prescription. The result is ℓ 1 I1 (x) I2 (y) =g ˆ2ℓ δI1I2 hOℓ Oℓ i (2π)2ℓ (x y)2ℓ Σ − I1 I2 I3 gˆ ℓ1ℓ2ℓ3 I1I2I3 1 ℓ (x) ℓ (y) ℓ (z) = t , hO 1 O 2 O 3 i N (2π)Σ (x y)2α3 (y z)2α1 (x z)2α2 − − − 5.4 Two- and three-point functions 167

I1I2I3 where Σ=ℓ1 + ℓ2 + ℓ3 and the tensor t is the unique invariant that can be con-

ℓ2+ℓ3 ℓ1 I2 I3 ℓ1+ℓ3 ℓ2 structed contracting α1 = − indices between t and t and α2 = − 2 i1...iℓ2 i1...iℓ3 2 ℓ1+ℓ2 ℓ3 and α3 = 2 − respectively between the other two pairs. After rescaling the CPO’s according to (2π)ℓ I ˜I = I , Oℓ −→ Oℓ gˆℓ√ℓ Oℓ the above correlators are found to agree perfectly with the free field calculation in =4 super Yang–Mills. N This result is expected to be valid only in the large N limit at fixed and largeg ˆ, when the Yang–Mills coupling gYM goes to zero. However in [147] it was proved that the one-loop corrections to the above correlators are zero. As a consequence the AdS calculation agrees with the exact perturbative contribution to two- and three-point functions of CPO’s for arbitrary gYM . This strongly suggests that the conjectured correspondence should be valid at finite N as well. In [147] a component formulation in terms of =1 multiplets is employed. The chiral primary operators that are N considered are of the form

tr Xℓ = tr ϕI1 ...ϕIℓ (5.34) ℓ † † † tr X = tr ϕI1 ...ϕIℓ , (5.35) I where ϕ and ϕI† , with I =1, 2, 3, are the complex scalar ﬁelds belonging to the chiral multiplets. The action of the model in this formulation was given in equation (2.11). The proof of the vanishing of the one-loop corrections to correlators of CPO’s given in [147] is based on a non-renormalization ‘theorem’ [165] for the two-point function

2 2 2 2 tr X tr X† = tr X tr X† h i and on clever colour combinatorics. The non-renormalization theorem says that the one loop correction to the above correlator vanishes. This is a consequence of the 2 2 following relations. The one-loop corrections to tr X tr X† come from the dia- h i grams

B(x, y) + = c (5.36) (x y)4 − 168 Chapter 5. AdS/SCFT correspondence

A(x, y) + = 2c (5.37) (x y)4 −

Notice that in the formulation of (2.11), that is used in [147], the Wess–Zumino gauge is exploited, so that the one loop corrections to the propagator, represented by the bubbles in (5.37), are not zero. Moreover the second term in (5.36) is the contribution of the ‘D-term’, while the ‘F -term’ does not enter. Both B(x, y) and A(x, y) are logarithmically divergent (see section 3.1), but one can prove the non- renormalization theorem

B(x, y)+2A(x, y)=0 . (5.38)

ℓ ℓ The one-loop correction to the two-point function tr X tr X† h i

ℓ ℓ tr X tr X†

comes from the sum of diagrams of the following types

ℓ ℓ tr X tr X†

ℓ ℓ tr X tr X† 5.4 Two- and three-point functions 169

ℓ ℓ tr X tr X†

One must consider the sum of contributions in which the ‘two-particle’ interaction structures (5.36) and (5.37) are inserted between each pair of lines. In [147] it is shown, through a combinatoric analysis, that this sum leads to a total one-loop correction of the form

k k [B(x, y)+2A(x, y)] tr X X† kN , h ∼ (x y)2k − which vanishes thanks to the non-renormalization theorem for the k=2 case. Analogously one can prove the vanishing of the one-loop correction to the three

ℓ1 ℓ2 ℓ3 point function tr X tr X tr X† h i tr Xℓ2 ℓ3 tr X† tr Xℓ1 To summarize the expressions of two- and three-point functions of CPO’s in perturbation theory are given by the free-field result and, as shown in [148], this coincides with what is found in the supergravity calculation in AdS5. As already remarked supersymmetry allows to generalize this result to all the composite operators of the current multiplet. In [147] this has been explicitly verified for a three-point function involving the operator AB of equation (2.18), which is a descendant 4 of CPO’s. E Notice that in this computation it is crucial to take into account the modification (2.19) of the operator AB in the non Abelian case [89]. E

4 AB is a descendant under superconformal transformations, but a primary ﬁeld from the view- pointE of conformal symmetry. 170 Chapter 5. AdS/SCFT correspondence

5.5 Four-point functions

Unlike two- and three-point functions which, as discussed in the previous section, are completely determined by superconformal symmetry up to constants, four-point functions are only determined up to an arbitrary function of two independent cross- ratios, for instance those introduced in section 4.5

2 2 2 2 x12x34 x14x23 r = 2 2 , s = 2 2 . (5.39) x13x24 x13x24 As a consequence the comparison of four-point amplitudes computed in supergrav-

ity/superstring theory in AdS5 with super Yang–Mills correlation functions would represent a truly ‘dynamical’ and non-trivial check of the correspondence. Such a check has been carried out, in an indirect way, for the non-perturbative correction to a four-point function in [89] using the exact match of a correlation function of sixteen fermionic operators in the same supersymmetry multiplet and will be reviewed in the next section. The calculation of four-point Green functions of composite operators in =4 su- N persymmetric Yang–Mills is rather complicated. The application of =1 superspace N techniques to this problem will be examined later. On the supergravity/superstring side the computation of four- and higher-point amplitudes with the approach of section 5.4 presents a new diﬃculty, with respect to the case of two- and three–point functions, related to the existence of two kinds of contributions, associated with “contact” and “exchange” diagrams.

Figure 5.3: Contact 4-point am- Figure 5.4: Exchange 4-point amplitude plitude

The former, see ﬁgure 5.3, are completely analogous to the above discussed contributions to two- and three-point functions, while the latter correspond to diagrams like the one in ﬁgure 5.4 and, as a new feature, involve also bulk to bulk propagators in 5.5 Four-point functions 171

AdS5. The diagram of ﬁgure 5.4 represents an s-channel amplitude, analogously one must take into account t and u channels. Moreover one must consider the possible exchange of all the states, for ﬁnite N also string states, coupled to the external vertices in the supergravity/superstring action. The corresponding amplitude has the form

4 4 d x′dρ′ d x′′dρ′′ A(x , x , x , x )= [K(x′, ρ′; x , 0) 1 2 3 4 ρ 5 ρ 5 1 · Z ′ ′′ K(x′, ρ′; x , 0)G(x′, ρ′; x′′, ρ′′)K(x′′, ρ′′; x , 0)K(x′′, ρ′′; x , 0)] , · 2 3 4 where G(z′; z′′) denotes the bulk to bulk propagator and, for compactness of notation, indices and possible derivatives acting on the various lines, depending the explicit form of the trilinear couplings, have been suppressed. The ﬁrst computations of this kind have been presented in [166, 106] for scalar four-point functions. The amplitudes considered are

Aφφφφ(x1, x2, x3, x4)

Aφχφχ(x1, x2, x3, x4) (5.40)

Aχχχχ(x1, x2, x3, x4) , where φ and χ=C(0) are the the two massless scalars of the type IIB superstring, the dilaton and the axion respectively. The operator corresponding to φ is the “glueball ﬁeld” = tr(F 2), whereas χ corresponds to = tr F F˜ , so that the amplitudes G C (5.40) should give the Green functions

(x ) (x ) (x ) (x ) hG 1 G 2 G 3 G 4 i (x ) (x ) (x ) (x ) (5.41) hG 1 C 2 G 3 C 4 i (x ) (x ) (x ) (x ) hC 1 C 2 C 3 C 4 i in =4 super Yang–Mills. N The relevant part of the type IIB supergravity action in AdS5 is

1 1 1 S = d5z√g R (∂φ)2 e2φ (∂χ)2 . (5.42) 2κ2 − 2 − 2 0 Z Notice that since the scalar ﬁelds φ and χ correspond to zeroth-order spherical harmonics, one can forget about the S5 factor and consider φ = φ(z) and χ = χ(z) as depending only on the AdS5 coordinate. 172 Chapter 5. AdS/SCFT correspondence

In the case of the Aφφφφ amplitude the only exchange diagrams that contribute involve the graviton propagator and at this order there are no contact terms, in the

Aφχφχ amplitude both the graviton and the axion propagators enter, while Aχχχχ receives contribution from the exchange of gravitons and dilatons. Furthermore contact

diagrams contribute to Aφχφχ and Aχχχχ. The fundamental diﬃculty in the computation of the above amplitudes is the

construction of the propagators in the AdS5 space. An interesting result has been proved in [106]. It has been shown that the graphs corresponding to the χ-exchange in

the amplitude Aφχφχ actually coincide with the contact graph. This is a consequence

of the fact that the scalar bulk to bulk propagator G(z; z′) satisﬁes an equation of the form

∆ G(z; z′)= δ(z z′) z −

in AdS5. Integrating by parts in the exchange amplitude, one obtains exactly a factor

∆zG(z; z′) and hence one of the two integrations over AdS5 can be performed using the δ-function. However this result does not hold in general and so the knowledge of

the propagators for all the relevant ﬁelds in AdS5 is necessary.

The AdS5 propagator for scalar ﬁelds has been computed in [167], while the

propagator for gauge ﬁelds was given in [168]. The graviton propagator in AdS5 has been computed only very recently [169]. As already remarked in section 5.1 the type IIB supergravity action contains terms that receive also non-perturbative corrections from D-instantons. In particular D-instantons contribute to the 4 term (5.7) and to the Λ16 term (5.8). As argued in R [170, 89] the corresponding contributions to supergravity/superstring amplitudes in

AdS5 should be compared with instanton contributions to the correlation functions of the associated composite operators in =4 supersymmetric Yang–Mills theory. This N comparison has been achieved in [89] for the correlation functions presented in the previous chapter (see sections 4.5 and 4.6). The results of [89] will be reviewed in the following sections. Notice that for the non-perturbative contribution to scattering amplitudes, with the “minimal” number of insertions leading to a non-vanishing result, only contact terms are relevant 5. As a consequence it is easy to single out the contributions to be compared with the Yang–Mills computations. In particular notice that there is no ambiguity related to possible cancellations among contact and

5Two- and three-point functions do not receive non-perturbative corrections at lowest order. 5.5 Four-point functions 173

exchange integrals. Moreover in general the contact non-perturbative contribution to an amplitude has schematically the form d4xdρ K K ...K , ρ5 ∆1 ∆2 ∆n Z which exactly coincides with the form expected for an instanton contribution in =4 N Yang–Mills theory after the integrations over the fermionic collective coordinates have been performed, since the one-instanton moduli space is exactly AdS5 [89, 90, 91, 92] 2 and the instanton ‘proﬁle’ tr F µν is identical to the bulk to boundary propagator

K∆ for ∆ = 4. This implies that the comparison with the Yang–Mills calculation can be performed before computing the integrals. These considerations will be exploited in the next section, where specific examples will be considered. Some remarks on the computation of perturbative contributions to four-point Green functions will now be made. The calculation of correlation functions of composite operators in =4 super Yang–Mills theory can be simplified using superspace N techniques. Although the explicit calculations have not yet been completed the application of this method to chiral primary operators will be briefly sketched. The approach followed in [147] and reviewed in the previous section for the computation of three-point functions of CPO’s can be generalized to correlation functions of =1 superfields whose lowest components are chiral primary operators of the form N of equation (5.35). The superfields to be considered are of the form

I ℓ ℓ tr Φ , tr ΦI† , I =1, 2, 3 . (5.43) The simplest case, ℓ=2, will be discussed here. Exploiting the SU(3) symmetry, that is manifest in the =1 formulation, one can restrict to Green functions with ﬁxed N values of the indices I, which allows to simplify the calculation. The chiral primary ij ij operators with ℓ = 2 are the scalars in (2.18). The operators belong to the 20R Q Q of the SU(4) R-symmetry group, which decomposes with respect to the SU(3) U(1) × subgroup that is manifest in the =1 formulation according to N

20R 6 4 + 6 4 + 80 , (5.44) ⊃ − 3 3 where the subscript on the right hand side refers to the U(1) charge. The corresponding decomposition of ij reads Q IJ I J 6 = tr Φ Φ 6 , 6 = tr Φ† Φ† 6 , O ∈ O IJ I J ∈ I I I V δ J K V 8 = tr Φ e Φ† Φ e Φ† 8 . (5.45) O J J − 3 K ∈ 174 Chapter 5. AdS/SCFT correspondence

For the sake of simplicity the attention will be restricted to Green functions of oper- IJ ators 6 and 6 . O O IJ Hence one type of four-point functions that one must consider is of the form

(a) 2 I 2 2 I 2 G = tr Φ† tr Φ tr Φ† tr Φ , I h I I i

with ﬁxed I. There are two diﬀerent tree-level contributions, corresponding to the diagrams (the notations are the same as in chapter 3)

2 I 2 2 I 2 tr ΦI† tr Φ tr ΦI† tr Φ

I 2 2 I 2 2 tr Φ tr ΦI† tr Φ tr ΦI†

The connected diagram is proportional to N 2 while the disconnected one goes like N 4. The only non-vanishing corrections to this Green function come from the diagrams

2 I 2 2 I 2 tr ΦI† tr Φ tr ΦI† tr Φ

I 2 2 I 2 2 tr Φ tr ΦI† tr Φ tr ΦI†

plus the analogous diagrams in which the VV internal propagator is inserted between diﬀerent pairs of external lines. Notice in particular that the following correction, 5.5 Four-point functions 175

obtained from the disconnected diagram,

2 I 2 tr ΦI† tr Φ

I 2 2 tr Φ tr ΦI† is trivially zero because of colour contractions. Moreover working in the superﬁeld formalism and choosing the Fermi–Feynman gauge, there are no corrections to the propagators to be included. Also, because of the non-renormalization properties of two-point functions discussed in the previous section, there is no non-vanishing disconnected diagram contributing to the four-point function at ﬁrst or higher order.

A second kind of four-point functions is of the form

(b) 2 I 2 2 J 2 G = tr Φ† tr Φ tr Φ† tr Φ , IJ h I J i

with I = J. In this case the only tree-level diagram is disconnected 6

2 2 tr ΦI† tr ΦJ†

tr ΦI 2 tr ΦJ 2 and goes as N 4. There is a unique non vanishing ﬁrst order correction corresponding 176 Chapter 5. AdS/SCFT correspondence

to the connected diagram

2 2 tr ΦI† tr ΦJ†

tr ΦI 2 tr ΦJ 2

where the internal propagator corresponds to the exchange of a chiral superﬁeld with index K = I = J. Just like in the previous case there is no disconnected ﬁrst order 6 6 correction because of the non-renormalization properties of two-point functions. The previous diagrams can be evaluated using the techniques of chapter 3. Since the points where the composite operators are inserted are external and not interaction points the result will not be local in the fermionic variables of =1 superspace. This N is not in agreement with the results of a calculation presented in [171]. An analysis similar to the one described here has been carried out in [172] employing a formulation of the model in =2 harmonic superspace. In [172] four-point N functions are explicitly computed in terms of polylogarithm functions. The expressions obtained resemble the one that was obtained in section 4.5 from the instanton contribution to the correlator . hQQQQi It has been observed by various authors [166, 106, 109, 107, 108, 173, 174] that the four-point functions computed in the AdS/CFT correspondence develop logarithmic singularities, when the limit of two coincident points is considered. This kind of behaviour has already been pointed out in section 4.5. In all the papers [166, 106, 109, 107, 173, 174] the short-distance behaviour has been shown taking the limit in

the unintegrated expression obtained from the supergravity amplitude in AdS5. This is rather subtle because the exchange of limit and integration might be problematic, moreover, as observed in [106], there could be compensations among various AdS diagrams. However this kind of cancellations cannot take place in the cases considered in [107, 108]. In particular in [107] some non-perturbative contributions from D- instantons to the four-point functions of scalar operators corresponding the dilaton and axion ﬁelds has been considered. The results of section 4.5, where the instanton

contribution to the four-point function G 4 has been given explicitly, unconfutably Q show the logarithmic behaviour. A satisfactory explanation of the origin of these 5.6 D-instanton eﬀects 177

singularities as well as of the reason for the lack of the expected pole-type singularities has not been proposed yet. In [109] an interesting analysis relating the contribution of contact and exchange amplitudes in AdS to the OPE of the corresponding operators in =4 super Yang–Mills has been given, using the technique of ‘conformal partial N wave expansion’, but the possible cancellation of logarithms between the two types of contributions has not been excluded. From the point of view of the boundary conformal ﬁeld theory the central point appears to be the presence of anomalous dimensions for some operators that does not violate the conformal symmetry. This issue is under active investigation [110] and will be brieﬂy addressed in the concluding remarks. Notice however that the behaviour of the correlation function G 4 for coincident points does not violate the non- Q renormalization theorems for three-point functions of the previous section, so that such non-renormalization properties could be generalizable to the non-perturbative level, at least for protected operators. In fact the limit in which two or two op- Q Q erators are taken to coincide in G 4 , that would correspond to one of the three-point Q functions that are expected to be tree-level exact, gives a vanishing result thanks to the pre-factor in equation (4.43).

5.6 D-instanton eﬀects in d=10 type IIB superstring theory

As already discussed the low energy type IIB supergravity action contains a f (τ, τ) 4 4 R 16 term as well as a supersymmetry related f16(τ, τ)Λ term. Both of these terms receive contributions from D-instantons of the type IIB theory. The AdS/CFT correspondence described in the previous section allows to associate to these terms well defined contributions to correlation functions of composite operators in the boundary Yang–Mills theory. It is natural to expect that the D-instanton generated terms should be related to instanton effects in =4 super N Yang–Mills. The type IIB D-instantons are Dirichlet p = 1 branes, i.e. they are localized − objects in ten-dimensional space-time, which means that they have a zero-dimensional world-volume. The charge-K D-instanton in flat ten-dimensional space-time is a finite-action solution of the classical Euclidean equations of motion in which the φ metric (in the Einstein frame) is flat and the complex scalar, τ = χ + ie− , has a non-trivial profile with a singularity at the position of the D-instanton, whereas the 178 Chapter 5. AdS/SCFT correspondence

other ﬁelds vanish [175]. Moreover the D-instanton is a BPS saturated conﬁguration preserving half of the supersymmetries. In the following the constant expectation values of the dilaton and the axion in the zero instanton sector will be denoted by φ˜ andχ ˜ respectively, while hatted symbols will refer to the classical D-instanton solution. To compute the scalar part of the type IIB action in the D-instanton background it is useful to dualize the R R scalar, χ = C(0), to an eight-form potential, C(8). ⊗ Since the metric is trivial, the action (up to constants) can be put in the form

1 φ (8) φ (8) 1 φ (8) S = dφ e− dC dφ e− dC e− dφ dC = 4 ± ∗ ∧∗ ± ∗ ± 2 ∧ Z Z 1 φ (8) φ (8) 1 φ (8) = dφ e− dC dφ e− dC e− dC , (5.46) 4 ± ∗ ∧∗ ± ∗ ∓ 2 Z Z∂M where ∂M denotes the boundary of space-time, that in the presence of the D-instanton is understood as made of the sphere at infinity and an infinitesimal sphere around the location of the instanton. The first term in (5.46) is semi-positive definite, so that it follows that the action satisfies a Bogomol’nyi bound

1 φ (8) S e− dC . (5.47) ≥ 2 Z∂M

Minimizing the action yields a BPS condition relating φ and C(8), which, expressed in terms of χ, becomes [175]

φ ∂ χ = i∂ e− . (5.48) Σ ± Σ The D-instanton solution is constructed through the ansatz

χ =χ ˜ + if(r) ,

where r = X X and f(r) 0, as r . After imposing (5.48) the equation | − 0| → → ∞ for the dilaton outside a sphere centered at the D-instanton location becomes

φ ΛΣ φ e− g e =0 . (5.49) ∇Λ∇Σ The solution is the harmonic function [175]

4 (10) 3Kα eφˆ = g + ′ . (5.50) s π4 X X 8 | − 0| 5.6 D-instanton eﬀects 179

Λ Λ where X is the ten-dimensional coordinate and X0 is the location of the D-instanton.

In (5.50) gs is the asymptotic value of the string coupling and the normalization of the second term has a quantized value [175] by virtue of a condition analogous to the Dirac–Nepomechie–Teitelboim condition that quantizes the charge of an electrically charged p-brane and of its magnetically charged p′-brane dual [176]. It is notable that the solution in (5.50) is simply the Green function for a scalar ﬁeld to propagate from X to X subject to the boundary condition that eφ = g at X or X . 0 s | |→∞ | 0|→∞ The solution for χ follows from (5.48) and is of the form

1 φˆ χ =χ ˜ + i A + e− , (5.51) − g s where A is an arbitrary constant. The action for a single D-instanton receives contribution only from the boundary term in (5.46), i.e. from the integration over the sphere at infinity and over an infinitesimal sphere around X0. The arbitrariness in the constant A can be used to simplify the computation. In particular with the choice A=0 in (5.51) the entire D- instanton action comes from the boundary of the infinitesimal sphere. Substituting for f from (5.51) gives 2π K SK = | | . (5.52) gs On the other hand, with the choice A = 1 in (5.51) the expression (5.46) reduces to gs an integral over the boundary at infinity, but the total action remains the same as

SK in (5.52). The bosonic zero-modes associated with the D-instanton solution are parameter- Λ ized by the coordinates X0 of the center of the D-instanton. As already remarked the D-instanton background breaks half of the supersymmetries. The fermionic zero- modes can be generated by acting with the broken supersymmetries on the scalar solution. The D-instanton contributions to the eﬀective action of type IIB supergravity can be deduced by studying scattering amplitudes in the instanton background. In this way one obtains in particular the corrections to the 4 and Λ16 terms in equations R (5.7) and (5.8) [114, 130]. In [170] it was pointed out that the 4 term vanishes when computed in the R AdS S5 background, because it involves a fourth power of the (vanishing) Weyl 5 × tensor. The ﬁrst, second and third functional derivatives of 4 vanish as well, and as R 180 Chapter 5. AdS/SCFT correspondence

a result one ﬁnds no corrections to zero, one, two and three-point amplitudes when the 4 term acts as source for Yang–Mills operators on the boundary. However, there R is a non-zero four-graviton amplitude arising from this term. The boundary values of these gravitons are sources for various bosonic components of the Yang–Mills current

supermultiplet. For example, the components of the metric in the AdS5 directions couple to the stress tensor, µν , whereas the traceless components polarized in the S5 T directions couple to massive Kaluza-Klein states. A linear combination of the trace of the metric on S5 and the fluctuation of the R R four-form potential couples to ⊗ the scalar components, ij. As a consequence the instanton contribution to the four- Q point correlation function G 4 computed in the previous chapter should be related Q to the appropriate D-instanton correction extracted from f (τ, τ) 4. However it is 4 R extremely laborious to single out the exact terms to be compared with the Yang–Mills computation, so that it will be convenient to show the matching of the D-instanton 16 correction to the Λ term with the GΛˆ 16 Green function of section 4.6. All the other terms of the same dimension, that are related to 4 by supersymme- R try can be picked out, for example, associating the physical fields with the components of an on-shell type IIB superfield [177]. As already remarked included among these is the sixteen-fermion interaction (5.8). Supersymmetry arguments [178, 179] imply

that the functions f16 and f4 are related by the action of SL(2,Z) modular covariant derivatives

f (τ, τ)=(τ )12f (τ, τ) , (5.53) 16 2D 4 where = ∂ +2i q is the covariant derivative that maps a (q,p) modular form into a D ∂τ τ2 (q+2,p) form (where the notation (q,p) labels the holomorphic and anti-holomorphic

SL(2,Z) weights of the form). Whereas f4 transforms with modular weight (0, 0), the function f has weight (12, 12) and therefore transforms with a non-trivial phase 16 − under SL(2,Z). This is precisely the phase required to compensate for the anomalous

U(1)B transformation of the 16 Λ’s, so that the full expression (5.8) is invariant under the SL(2,Z) S-duality group. In the small coupling limit, all of these terms can be expanded in the form

φ/2 2φ e− fn = anζ(3)e− + bn +

∞ ∞ φ n 7/2 2πiKτ K φ k + µK(Ke− ) − e 1+ ck,n(Ke− )− . (5.54) K=1 ! X Xk=1 The ﬁrst two terms have the form of string tree-level and one-loop contributions, 5.6 D-instanton eﬀects 181

while the last one has the appropriate weight of e2πiKτ to be associated with a charge- K D-instanton effect. Anti-D-instanton contributions have not been displayed. The K coefficients ck,n as well as the ‘D-instanton partition functions’ µK are explicitly given in [130]. The coefficient of the leading term in the series of perturbative fluctuations around a charge-K D-instanton is independent of which particular interaction term is being discussed and reads

φ 7/2 2πiKτ = µ (Ke− )− e . ZK K

It should be identiﬁed with the contribution of a charge-K D-instanton to the measure in string frame which, up to an overall numerical factor, reads

(s) 1 10 16 dΩ =(α′)− d Xd Θ . (5.55) K ZK

Actually there are subtleties in doing this, since the full series is not convergent, it is rather an asymptotic approximation to a Bessel function.

5.6.1 Testing the AdS/CFT correspondence: type IIB D- instantons vs. Yang–Mills instantons

As observed in [170] the charge-K D-instanton action that appears in the exponent in (5.6) coincides with the action of a charge-K Yang–Mills instanton in the boundary theory, which indicates a correspondence between these sources of non-perturbative eﬀects. This idea is reinforced by the correspondence between other factors. For ex- g2 ample, after substituting eφ = YM and α = L2N 1/2g 1 , the measure (5.55) contains 4π ′ − YM− 8 an overall factor of the coupling constant in the form gYM . Indeed, this is exactly the power expected on the basis of the AdS/CFT correspondence, since the one-instanton contribution to the Green functions in the = 4 Yang–Mills theory, considered in N 8 chapter 4, also has a factor of gYM arising from the combination of the bosonic and fermionic zero-mode norms. In this section the leading instanton contributions to type IIB superstring amplitudes will be compared with the corresponding =4 cur- N rent correlators considered in chapter 4. In the next section the same analysis will be carried out studying the semiclassical ﬂuctuations around the AdS S5 D-instanton 5 × solution. To compare the Yang–Mills instanton calculation with an amplitude obtained according to Maldacena’s prescription one must expand the function f16(τ, τ) in (5.53) 182 Chapter 5. AdS/SCFT correspondence

to extract the one-instanton term. In order to compare with the Yang–Mills sixteen- point correlator one needs to consider the situation in which all sixteen fermions propagate to well defined configurations on the boundary. Equation (5.28) leads to the normalized bulk-to-boundary propagator of the fer- 3 ˆ mionic field Λ of AdS ‘mass’ m = 2L , associated to the composite operator Λ of 7 − dimension ∆ = 2 , see equation (2.18),

F 1 µ K7/2(ρ0, x0; x)= K4(ρ0, x0; x) (ρ0γ5ˆ +(x0 x) γµˆ) . (5.56) √ρ0 − Suppressing all spinor indices, one obtains

4 F ΛJ (x0, ρ0)= d xK7/2(ρ0, x0; x)JΛ(x) , (5.57) Z

where JΛ(x) is a left-handed boundary value of Λ and acts as the source for the composite operator Λˆ in the boundary =4 Yang–Mills theory. As a result, the N classical action for the operator (Λ)16 in the type IIB supergravity on AdS S5 is 5 × 4 2π( 1 +iχ) 12 d x0dρ0 S [J]= e− gs g− V 5 Λ s S ρ5 Z 0 16 1 t K (ρ , x ; x ) ρ γ5ˆ +(x x )µγ J (x ) , (5.58) 16 4 0 0 p ρ 0 0 − p µˆ Λ p p=1 √ 0 Y φ where e = gs and χ =χ ˜ (since the scalar ﬁelds are taken to be constant in the 5 3 5 AdS S background) and V 5 = π is the S volume. The 16-index invariant 5 × S tensor t16 is the same as the one deﬁned after (4.55). The overall power of the coupling 25 φ 25 1 2 constant comes from the factor of (e− )− 2 = in the expansion (5.54) and the gs 1 16 factor of α′ in front of the Λ term in the type IIB action, which using the relation (5.13) yields

25 12 1 1 2 1 , α g ∼ g ′ s s where an overall numerical constant has been dropped. Using the dictionary (5.15) and diﬀerentiating with respect to the chiral sources this result agrees with the expression (4.55) of section 4.6 obtained in the Yang Mills calculation. In particular

the overall power of gYM in (4.55) is reproduced after introducing the same rescaling

of each Λ in (5.58) by a factor of 4πgs as in equation (4.53). Actually the overall numerical constant has not been checked, see however [92]. Notice that to match 5.6 D-instanton eﬀects 183

the result of the Yang–Mills calculation it is crucial that in the product of sixteen F 5 (x x0)µ µ matrices K exactly eight factors of ρ γ and eight factors of − γ are picked 7/2 √ 0 √ρ0 out. This is ensured by the contraction with t16, which selects the correct number of factors of each kind. In the next section this expression will also be motivated directly by semi-classical quantization around a D-instanton field configuration that is a Euclidean solution of the type IIB supergravity in AdS S5. 5 × The agreement of the Λ16 amplitude with the corresponding sixteen-current correlation function of chapter 4 is sufficient to guarantee that the instanton contributions to all the other Yang–Mills correlation functions, that are related to this term by =4 supersymmetry, will also agree with their type IIB superstring counterparts. N For example, the correlation function that has been considered in most detail in chap- ij ter 4 was the one with four superconformal scalar currents which are in the 20R Q of SU(4), and have dimension ∆=2 and AdS ‘mass’ m2 = 4 . As already discussed − L2 the supergravity field, Qij, that couples to ij is a linear combination of the fluc- Q 5 m tuation of the trace of the metric on S , hm , and of the four-form field potential, C(4) = ǫ Rf. Therefore, contributions to the correlation of four of these MNPQ MNPQR∇ composite scalars in the one D-instanton background should correspond to the leading parts of the K=1 terms in the expansion of the 4 interaction (5.7) as well as 2 4 R terms of the form 2 F (5) and F (5) . These last two terms involve F (5), the R ∇ ∇ self-dual field strength of the antisymmetric four-form potential, and are related by supersymmetry to the 4 term. R It follows from the structure of (4.40) that the Yang–Mills instanton contribution to each factor of ij is of the form K (xµ, ρ ; xµ, 0), where the two derivatives Q ∇∇ 2 0 0 are not necessarily contracted. But this is the expected form for a propagator from the AdS S5 bulk to the boundary for a scalar field of dimension 2. Therefore, 5 × at least the general form of the expression obtained from the K=1 terms in the expansion of 4 and the related F (5) interactions agrees with the four- ij correlation R Q function in a Yang–Mills instanton background. In order to see this agreement in more 2 detail it would first be necessary to determine the precise form of the F (5) and 4 ∇ F (5) interactions that contain the fluctuations of F (5). The K=1 contribution to ∇ m the amplitude of four fluctuations of the appropriate combination of hm and f can then be extracted from these interaction terms. Although this computation has not been performed explicitly, the result is guaranteed to reproduce the K=1 expression obtained from the Yang–Mills theory in chapter 4 since it is related by supersymmetry to the Λ16 amplitude. 184 Chapter 5. AdS/SCFT correspondence

The analogous comparison of the correlation function of eight AB’s with the am- E plitude for eight EAB’s in the type IIB superstring theory proceeds in much the same way. The supergravity fields, EAB, that couple to AB arise from the internal compo- E nents of the complex type IIB antisymmetric tensor. Supersymmetry relates an H8 term, where H is the complex combination of the type IIB three-form field-strengths, H and H , introduced in (5.6), to the 4 term in the type IIB effective action. One 1 2 R thus expects interactions schematically of the form ( K )8, with various contrac- ∇ 3 tions of the derivatives. Using the explicit form of the spinor collective coordinates A A µ ¯A AB ζ = (ρ0η +(x x0)µσ ξ )/√ρ0 one may check that the gaugino bilinears − E in the one-instanton background exactly give rise to K3 and derivatives thereof, as expected for the scalar propagator of a scalar field of dimension ∆=3 and AdS mass m2 = 1 . Although a precise matching of the resulting amplitudes has not been − L2 performed, one may appeal to supersymmetry arguments to determine the complete structure of these terms. Notice that the above considered non-perturbative terms in the type IIB supergravity effective action when expanded around the AdS S5 background give 5 × rise to both derivative and non-derivative interaction terms. The matching of the non-derivative “mass-related” terms considered here with corresponding terms in the Yang–Mills Green functions is rather straightforward, but clearly it is only a hint to the conjectured correspondence.

5.7 The D-instanton solution in AdS S5 5 × 1 The results of the previous section will now be reinforced by studying how the (α′)− terms in the type IIB eﬀective action can be determined by semi-classical type IIB supergravity ﬁeld theory calculation in a D-instanton background. The solution of the equations of motion of the type IIB theory in Euclidean AdS S5 will be now discussed. The BPS condition for a D-instanton in this 5 × background is identical to (5.48) and together with the equation of motion for the dilaton leads to

gΛΣ eφ =0 (5.59) ∇Λ∇Σ and determines the complex scalar τ. In the Einstein frame the Einstein equations are unaltered by the presence of the D-instanton, because the associated Euclidean stress energy tensor vanishes. As a consequence AdS S5 remains a solution. Equation 5 × 5 5.7 AdS5 S D-instanton solution 185 ×

(5.59) is identical to the equation for the Green function of a massless scalar propa- µ i µ i gating between the location of the D-instanton (x0 ,y0) and the point (x ,y ), which is the bulk-to-bulk propagator (subject to the boundary condition that it is constant in the limits ρ 0 and ρ ). This is easy to solve using the conformal ﬂatness of → →∞ 5 Λ µ i 2 2 2 AdS S . In Cartesian coordinates X =(x ,y ), (5.14) reads ds = L ρ− dX dX, 5 × · where ρ2 = y y and the solution of (5.59) with a constant asymptotic behavior is of · the form

4 4 ˆ ρ ρ ˆ(10) eφ = g + 0 eφ g , (5.60) s L8 − s where ρ = y and eφˆ(10) is the harmonic function that appeared in the ﬂat ten- 0 | 0| dimensional case, (5.50). In evaluating D-instanton dominated amplitudes one is only interested in the case in which the point (xµ,yi) approaches the boundary (ρ y ≡| |→ 0), in which case it is necessary to rescale the dilaton proﬁle (just as it is necessary to rescale the scalar bulk-to-bulk propagator, [118, 119]), so that the combination

4 4 4 φˆ 3K(α′) ρ0 − ρ e gs = 8 4 2 2 4 , (5.61) − L π ((x x0) + ρ0) − will be of relevance in the ρ 0 limit. → As mentioned earlier, the correspondence with the Yang–Mills instanton follows from the fact that ρ4/((x x )2 + ρ2)4 = K is proportional to the instanton number 0 − 0 0 4 density, (F )2, in the =4 Yang–Mills theory. Strikingly, the scale size of the Yang– µν N Mills instanton is replaced by the distance ρ0 of the D-instanton from the boundary. This is another indication of how the geometry of the Yang–Mills theory is encoded in the type IIB superstring. Note, in particular, that as the D-instanton approaches the 4 φˆ boundary ρ 0, the expression for ρ− e reduces to a δ function that corresponds 0 → to a zero-size Yang–Mills instanton. The BPS condition implies that one can write the solution for the R R scalar ⊗ as

χˆ =χ ˜ + if(x, y) , (5.62)

θYM whereχ ˜ is the constant real part of the ﬁeld (which corresponds to 2π , see equation (5.15)) and just like in the ﬂat ten-dimensional case

1 φˆ f = A + e− . (5.63) − gs 186 Chapter 5. AdS/SCFT correspondence

Since the action is independent of constant shifts of χ, actually it does not depend on the arbitrary constant, A. In a manner that follows closely the flat ten-dimensional case considered in the appendix of [114] and reviewed in the preceding section, the action for a charge K D-instanton can be written as L10 dρd4xd5ω S = gΛΣ (e2φˆf∂ f) , (5.64) K −(α )4 ρ5 ∇Λ Σ ′ Z which reduces to an integral over the boundaries of AdS S5 and the surface of an 5 × infinitesimal sphere centered on the D-instanton at x = x0, y = y0. The result is exactly the same as in the flat case 2π K SK = | | . (5.65) gs By choosing A = 1 in (5.63) the expression (5.64) reduces to an integral over the gs boundary at ρ = 0. Remarkably, in this case the boundary integrand is identical to the action density of the standard four-dimensional Yang–Mills instanton. Whereas the AdS S5 metric remains unchanged by the presence of the D- 5 × instanton in the Einstein frame, it is radically altered in the string frame where the instanton is manifested as a space-time wormhole (as in the flat ten-dimensional case [175]). For finite values of K the dilaton becomes large in the Planck-scale neck and the classical solution is not reliable in that region. However, for very large instanton number, the neck region becomes much larger than the Planck scale so, by analogy with the D-brane examples studied in [38], it should be very interesting to study the implications of the modified AdS S5 geometry in the large-K limit of the large-N 5 × theory. The D-instanton contribution to the amplitude with sixteen external dilatinos, A Λα , may now be obtained directly by semi-classical quantization around the classical D-instanton solution in AdS S5. The leading instanton contribution can be deter- 5 × mined by applying supersymmetry transformations to the scalar field which has an instanton profile given by (5.61). Since the D-instanton background breaks half the supersymmetries the relevant transformations are those in which the supersymmetry parameter corresponds to the Killing spinors for the sixteen broken supersymmetries. 1 These Killing spinors have U(1)B charge 2 and are defined by a modified version of (5.17) that includes the non-trivial composite U(1)B connection, QM [126], that is made from the type IIB scalar field [175] i 1 ζ D Q ζ = γ ζ . (5.66) DM ≡ M − 2 M 2L M 5 5.7 AdS5 S D-instanton solution 187 ×

Substituting the Euclidean D-instanton solution into the expression for the composite connection gives

i φˆ φˆ Q = e− ∂ e , (5.67) M 2 M with φˆ deﬁned by (5.61). The solution of (5.66) is

Mˆ φˆ zM γ (0) ζ = e− 4 ζ , (5.68) ± √ρ0 ±

(0) (0) (0) where ζ is a constant spinor satisfying γ5ζ = ζ . ± ± ± ± The sixteen broken supersymmetry transformations associated with ζ(0) give rise − to the dilatino zero-modes,

M Λ(0) = δΛ=(γ PˆM )ζ , (5.69) − i where PˆM is the expression for PM ∂M τ ∗ in the D-instanton background [114], ≡ 2τ2 i.e.

φˆ φˆ PˆM = e− ∂M e . (5.70)

Using the Killing spinor equation and the D-instanton equation M Pˆ = 0 it is easy D M to check (recalling that PM has U(1)B charge 2) that 3 γM Λ = Λ , (5.71) DM (0) −2L (0) so that Λ(0) is a solution of the appropriate massive Dirac equation. The relevant amplitudes are those with external states located on the boundary, in which case one may use the fact that for ρ 0 ∼ 1 φˆ PM ∂M e (5.72) ∼ gs in (5.69), which leads to

4 φˆ Λ(0) (e gs)ζ . (5.73) ∼ gs − − This means that near ρ = 0 the dilatino proﬁle in the D-instanton background is 4 proportional to ρ K4(x0, ρ0; x, 0). As a result the leading contribution to the sixteen-dilatino amplitude again repro- duces the corresponding sixteen-current correlator in = 4 supersymmetric Yang– N Mills theory. Explicitly, the D-instanton approximation to the amplitude with sixteen 188 Chapter 5. AdS/SCFT correspondence

A external dilatinos, Λα , at points on the ρ = 0 boundary is (up to an overall constant factor)

16 4 1 (0) Ap 12 2πK( +iC ) d x0dρ0 16 (0) − gs 5 Λαp (xp, 0) J = gs− e VS 5 d ζ h i ρ − p=1 0 Y Z Z 16 1 α˙ pAp K (x , ρ ; x ) ρ ηAp +(x x ) σµ ξ J (x ) , (5.74) · 4 0 0 p ρ 0 αp p − 0 µ αpα˙ p Λ p p=1 √ 0 Y

where JΛ(xp) is the wave-function of the dilatino evaluated at the boundary point (0) ¯ (xp, 0) and the Grassmann spinor ζ is deﬁned in terms of η and ξ by − 5 M (0) 1 γ z γM ζ ζ(ρ0, x x0)= − , − 2 √ρ 0 where

ηα (0) M µ µ ζ = , z =(x x0 , ρ0) .  α˙  − ξ   12 The power of gs− has been inserted in (5.74) from the expression given in section (5.1) although this power should also follow directly by considering the normalization of the bosonic and fermionic zero modes. Up to the overall constant factor, the amplitude (5.74) agrees with (5.58) and therefore with (4.55). In similar manner the instanton proﬁles of all the ﬁelds in the supergravity mul-

tiplet follow by applying the broken supersymmetries to PM the appropriate number of times, just as in the ﬂat ten-dimensional case [114]. The single D-instanton contributions to any correlation function can then be determined. This is guaranteed to agree with the corresponding term in the expansion of the eﬀective type IIB action as well as with the corresponding =4 Yang–Mills correlation function. N Conclusions

This dissertation reports the study of various aspects of =4 supersymmetric Yang– N Mills theory. The perturbative finiteness properties of the theory have been known for a long time. However the analysis presented here allows to point out several problems in the perturbative computation of Green functions of elementary (super) fields. The calculations reviewed in chapter 3 show that the gauge-fixing procedure is quite problematic and presents subtleties both in the description in terms of component fields and using the =1 superfield formalism. The appearance of UV divergences in N off-shell propagators, when the Wess–Zumino gauge in components is exploited, has been discussed and it has been shown that relaxing the choice of the WZ gauge such divergences are cancelled. Different difficulties have been encountered in superspace computations: the supersymmetric generalization of the Fermi–Feynman gauge appears to be privileged since any different choice of gauge leads to IR divergences in off-shell Green functions. These issues have been studied in a systematic way in chapter 3. The divergences found in one-loop calculations are expected to be gauge artifacts, and it would be interesting to obtain an explicit check of the cancellation of such divergences in gauge invariant Green functions. As briefly discussed in the final chapter in the context of the AdS/SCFT correspondence, superspace techniques can be applied to the calculation of correlation functions of gauge-invariant composite operators as well. =1 N superfields should prove a powerful tool in this computations like in the case of elementary fields. At the perturbative level the effect of the inclusion of mass terms for the (anti) chiral superfields has also been discussed. The considered mass terms break super-

189 190 Conclusions

symmetry down to =1, but it is explicitly proved, at the one loop-level, that the N ultraviolet properties of the model are not affected by this modification. The same result had been obtained long ago by means of dimensional arguments and is reinforced by the analysis presented here. The possibility of employing this mass deformed =4 model as a supersymmetry N preserving regularization scheme for =1 theories has been recently proposed. As N observed in chapter 3 this approach should be generalizable to =2 theories. N Non perturbative effects in =4 supersymmetric Yang–Mills theory are studied in N chapter 4 with the aid of instanton calculus. The literature on =4 super Yang–Mills N contains examples of instanton computations of corrections to the effective action in the Abelian Coulomb phase. The calculations presented in chapter 4 have been performed in the superconformal phase, corresponding to the origin of the moduli space, which is related to type IIB superstring theory in AdS S5 by the duality 5 × recently proposed by Maldacena. In particular an exact field theoretical computation of the one-instanton contribution (in the semiclassical approximation) to a four-point function of scalar composite operators belonging to the current supermultiplet has been reported. The non-perturbative corrections to Green functions in =4 supersymmetric N Yang–Mills theory have been subsequently reconsidered in the context of the AdS/CFT correspondence. Previous checks of the conjectured correspondence had been obtained for two- and three-point functions at the perturbative level. The results reviewed here strongly support the possibility of extending the correspondence at the non-perturbative level as well. The agreement found between the contributions to four- and higher-point functions due to two different sources of non-perturbative effects, instantons in =4 super Yang–Mills and D-instantons in type IIB superstring N theory, represents a highly non trivial check of the conjecture. Furthermore these results have been obtained in the case of a SU(2) gauge group and therefore suggest that the correspondence, originally formulated for a SU(N) gauge group in the large N limit, may actually hold at finite N as well. Perturbative checks of this claim have already been proposed, but the non-perturbative results discussed here bring stronger evidence in this direction, since they concern four- and higher-point functions that are not fixed by superconformal symmetry. Despite these interesting results on the field theory side, there is still very little understanding of the correct way of dealing with string corrections to the supergravity approximation for finite N. This is a very interesting subject for future developments. In particular generalizations of the orig- Conclusions 191

inal conjecture to theories with less supersymmetry and even to non-supersymmetric theories have been proposed. In this context computations of quantities of phenomenological interest, such as glueball spectra, have been performed. However the possibility of comparing this results with, for instance, the continuum extrapolations from lattice simulations requires the extension of the correspondence to weak ’t Hooft coupling, i.e. to finite N including string effects. The four-point correlation function computed in chapter 4 displays logarithmic singularities when any two insertion points are taken to coincide. This peculiar behaviour in the operator product expansion (OPE) has also been observed by other authors in supergravity calculations in AdS5 and is a very interesting feature of the superconformal field theory at hand. In understanding this phenomenon a crucial rôle is played by the fact that the chiral primary operators, ∆, that enter the Green functions under consideration Oq are protected, i.e. their dimension ∆ and charge q are related by ∆ = q. The first point to observe is the absence of pole-type singularities in the OPE. This is a direct consequence of the condition ∆ = q. If one considers the OPE for chiral primary operators with charges q1 and q2,

(x) (y) for x y 0 , O(q1) O(q2) | − |→ there are two possibilities. Either the OPE contains only chiral primary operators and one has x + y (x) (y) c ∆ + terms vanishing as x y 0 , O(q1) O(q2) ∼ O(q1+q2) 2 | − |→ with ∆ = q1 + q2 and c a numerical constant, or there is a mixing with a non chiral primary operator, whose dimension is strictly larger than the charge, and then x + y (x) (y) ∆ + ... , O(q1) O(q2) ∼ Vq 2 where q = q1 + q2 and ∆ = q1 + q2 + γ > q1 + q2. As a consequence in any case there cannot be a pole-type singularity. The second situation described here could be responsible for the logarithmic behaviour. The anomalous dimension γ is determined by the calculation of the three point function , that is obtained by taking the hVOOi limit of coincident points in a four-point function of chiral primary operators like the one that is computed in chapter 4. Notice that one cannot study the limit in a three- point function, since three-point functions of chiral primaries are not renormalized. 192 Conclusions

The logarithmically divergent coeﬃcient in the OPE of and could originate O(q1) O(q2) from a perturbative expansion of γ = γ(g). More precisely if the operator has an V anomalous dimension, γ, in perturbation theory the latter is expected to be small and to admit an expansion in the coupling constant g. Expanding (x) (y) O(q1) O(q2) in powers of γ yields

1 (x) (y) 1 γ log(µ2(x y)2)+ hO(q1) O(q2) i ∼ (x y)2∆ − − − 1 + γ2(log(µ2(x y)2))2 + . . . , 2 − where an arbitrary mass scale µ has been introduced for dimensional reasons. At each non-trivial order in g, the last equation seems to violate conformal invariance. The exact expression is however conformally invariant by construction. One can reconcile the two results by observing that because of the presence of an anomalous dimension, the operator has to be renormalized at each order in g and as a consequence it does not provide a linear representation of the conformal group. Appendix A

Notations and spinor algebra

Minkowskian metric 1000 − 0 100 η = . (A.1) µν  0 010   0 001      σ-matrices

σµ =( 1I , σi) , σµ =( 1I , σi) , i =1, 2, 3 , (A.2) − 2 − 2 − where 0 1 0 i 1 0 σ1 = , σ2 = , σ3 = . (A.3) 1 0 i −0 0 1 − Weyl spinors are denoted as follows

α˙ ψα , ψ , with α, α˙ =1, 2 . (A.4)

Spinor indices are raised and lowered by the antisymmetric tensors εαβ (ε12 = ε21 = − α˙ β˙ 1˙2˙ 2˙1˙ ε = ε = 1) and ε (ε = ε = ε˙ ˙ = ε˙ ˙ = 1) − 12 21 − − 12 21 α αβ β ψ = ε ψβ , ψα = εαβψ , ˙ α˙ α˙ β˙ β ψ = ε ψβ˙ , ψα˙ = εα˙ β˙ ψ . (A.5)

The matrices σµ have the following index structure

µ µ σ = σ αα˙ . (A.6)

193 194 Appendix A. Notations and Algebra

Indices of the σµ matrices are raised and lowered by the ε tensor as well

µ αα˙ α˙ β˙ αβ µ σ = ε ε σ ββ˙ . (A.7)

Properties of the σ-matrices.

The following properties of the σ-matrices can be easily derived from their deﬁ- nition

(σµσν + σνσµ) β = 2ηµν δ β α − α µ ν ν µ α˙ µν α˙ (σ σ + σ σ ) ˙ = 2η δ ˙ , (A.8) β − β as well as the completeness relations

tr(σµσν) = 2ηµν − ˙ ˙ σµ σββ = 2δ βδ β . (A.9) αα˙ µ − α α˙ The generators of Lorentz transformations read 1 σµν β = σµ σν αβ˙ σν σµ αβ˙ α 4 αα˙ − αα˙ µνα˙ 1 µ αα˙ ν ν αα˙ µ σ ˙ = σ σ σ σ . (A.10) β 4 αβ˙ − αβ˙

Euclidean σ-matrices.

They are deﬁned as 1

σµ = (1I , iσi) , σµ = (1I , iσi) , i =1, 2, 3 . (A.11) E 2 E 2 − The Euclidean σ-matrices satisfy properties analogous to their Minkowskian couter- parts. In particular

µ ν ν µ β µν β (σEσE + σEσE)α = 2δ δα µ ν ν µ α˙ µν α˙ (σEσE + σEσE) β˙ = 2δ δ β˙ , (A.12)

µ ν µν tr(σEσE) = 2δ µ ββ˙ β β˙ σE αα˙ σE µ = 2δα δα˙ . (A.13) 1The subscript E is suppressed in the text. Notations and Algebra 195

The Euclidean Lorentz generators are 1 σµν β = σµ σν αβ˙ σν σµ αβ˙ E α 4 E αα˙ E − E αα˙ E µν α˙ 1 µ αα˙ ν ν αα˙ µ σ ˙ = σ σ σ σ . (A.14) E β 4 E E αβ˙ − E E αβ˙ The following realization of the Dirac γ-matrices is used

0 σµ 1 0 γµ = , γ5 = iγ0γ1γ2γ3 = . (A.15) σµ 0 0 1 −

Spinor algebra.

Contractions between spinor indices are taken according to the conventions

α α ψχ = ψ χα = ψαχ = χψ − α˙ ψχ = ψ χα˙ = ψ χ = χψ . (A.16) α˙ − α˙ Fierz rearrangements 1 1 ψαψβ = εαβψψ, ψ ψ = ε ψψ −2 α β 2 αβ ˙ α˙ β 1 α˙ β˙ 1 ψ ψ = ε ψψ , ψ ψ ˙ = ε ˙ ψψ . (A.17) 2 α˙ β −2 α˙ β Miscellaneous relations 1 θσµθθσνθ = θθθθηµν −2 1 1 (θψ)(θχ)= (θθ)(ψχ) , (θψ)(θχ)= (θθ)(ψχ) −2 −2 µ µ µ µ χσ ψ = ψσ χ , (χσ ψ)† = ψσ χ − µ ν ν µ µ ν ν µ χσ σ ψ = ψσ σ χ , (χσ σ ψ)† = ψσ σ χ 1 (ψχ)λ = (χσµλ)(ψσ ) . (A.18) α˙ −2 µ α˙

Superspace covariant derivatives.

Covariant derivatives in Minkowskian superspace are deﬁned as

∂ µ α˙ ∂ α µ Dα = + iσαα˙ θ ∂µ , Dα˙ = α˙ iθ σαα˙ ∂µ (A.19) ∂θα −∂θ − 196 Appendix A. Notations and Algebra

and satisfy

D , D = 2iσµ ∂ (A.20) { α α˙ } − αα˙ µ D ,D = D , D ˙ =0 . { α β} { α˙ β} Moreover

∂ ∂ ∂ ∂ ˙ ∂ ∂ εαβ = , εαβ θθ = εα˙ β θθ =4 . (A.21) α α β α˙ β˙ ∂θ −∂θα ∂θ ∂θ ∂θ ∂θ Other properties

β˙ µ α˙ [D , D ˙ ,D ]=0 , [D , D ˙ D ]= 4iσ ∂ D α˙ { β γ} α β − αα˙ µ 2 α˙ α˙ [D2, D ]= 8iDασµ D ∂ 162 =8iD σµ Dα∂ + 162 . (A.22) − αα˙ µ − αα˙ µ The corresponding Euclidean superspace derivatives read

E ∂ µ α˙ E ∂ α µ Dα = + σαα˙ θ ∂µ , Dα˙ = α˙ θ σαα˙ ∂µ (A.23) ∂θα −∂θ − and possess analogous properties

E DE, D = 2σµ ∂ (A.24) { α α˙ } − αα˙ µ E E E E D ,D = D , D ˙ =0 . { α β } { α˙ β }

E E E [D , D ˙ ,D ]=0 α˙ { β γ } ˙ E E E β µ Eα ˙ α E E µ α [Dα , Dβ˙ D ]= 4σE αα˙ ∂µD , [DEDα , Dα˙ ]= 4σE αα˙ ∂µDE 2 − α˙ α˙ − [D2 , D ]= 8Dασµ D ∂ 162 =8D σµ Dα∂ + 162 . (A.25) E E − E E αα˙ E µ − E E αα˙ E µ The following relations are useful in superspace calculations

2 2 2 2 2 2 2 2 2 2 DEDEDE = 16 DE , DEDEDE = 16 DE 2 2 α˙ β˙ D DEDED = 16σµ σν ∂ ∂ D D E α β E − E αα˙ E ββ˙ µ ν E E 2 E E 2 µ ν α β D D D ˙ D = 16σ σ ∂ ∂ D D . (A.26) E α˙ β E − E αα˙ E ββ˙ µ ν E E

Berezin integration.

Integration over a Grassmannian variable η is deﬁned as

dη =0 , dη η =1 . (A.27) Z Z Notations and Algebra 197

The power expansion of a function f(η) is simply

f(η)= a + bη , (A.28) so that

dη f(η)= b , dη ηf(η)= a . (A.29) Z Z Integration by parts in Berezin integrals is allowed since ∂ dη f(η)=0 . (A.30) ∂η Z The δ-function is deﬁned by

dη f(η)δ(η)= f(0) , (A.31) Z which implies δ(η)= η. The integration measure over the Grassmannian variables of =1 superspace is deﬁned as follows N d4θ = d2θd2θ 2 1 α β 2 1 α˙ β˙ d θ = ε dθ dθ , d θ = ε dθ dθ ˙ , (A.32) 4 αβ 4 α˙ β so that

d2θ θθ = d2θ θθ =1 . (A.33) Z Z Other useful formulas can be found in the appendices of [4, 3] Appendix B

Electric-magnetic duality

B.1 Electric-magnetic duality and the Dirac monopole

The free Maxwell equations ~ E~ =0 ~ B~ =0 ∇· ∂E ∇· ∂B (B.1) ~ B~ =0 ~ E~ + =0 , ∇ × − ∂t ∇ × ∂t are invariant under a duality transformation D, exchanging electric and magnetic ﬁelds according to

D : E~ B~ , B~ E~ . −→ −→ − More generally the equations are invariant under a continuous transformation E~ cos θ sin θ E~ . (B.2) B~ −→ sin θ cos θ B~ − A covariant formulation can be given using the language of diﬀerential forms. The free equations can be written

1 d F =0 , − d F =0 , ∗ ∗ where ( F ) = 1 ε F ρσ and d denotes the external derivative. The duality trans- ∗ µν 2 µνρσ formation simply becomes F F . In the presence of electric sources the equations → ∗ become

1 dF =0 , − d F = j , ∗ ∗ 198 B.1 Electric-magnetic duality 199

µ ~ where j is a one-form of components je =(ρe, je), so that the symmetry is broken. To recover the duality symmetry magnetic source terms must be included symmetrizing the equations as follows

1 dF = j , − d F = j . m ∗ ∗ e The system is then symmetric under the simultaneous exchange of F with F and of ∗ je with jm. Even if the inclusion of magnetic point-like particles in classical electromagnetism seems to be ruled out by the experimental results, it allows to derive a consequence that is of fundamental importance in the more interesting case of non Abelian theories, the Dirac quantization condition. A consistent quantization 1 of a theory with both electric and magnetic charges qi and gj requires that they satisfy the condition

q g =2πℏn , (B.3) i · j ij where nij are arbitrary integers. The quantum mechanical description of a charged particle in an electromagnetic

ﬁeld requires the introduction of the vector potential Aµ, related to the ﬁeld strength by F = ∂ A ∂ A , through the substitution ~ (~ iq A~). This leads to the µν µ ν − ν µ ∇→ ∇ − ℏ minimal coupling in the Schroedinger equation

∂ψ ℏ2 iq iℏ = (~ A~)2ψ + V ψ . ∂t −2m ∇ − ℏ Gauge transformations for this system read

i q χ(θ) ψ e− ℏ ψ , A~ A~ ~ χ(θ) . (B.4) −→ −→ − ∇ In the presence of a point like magnetic charge it is not possible to associate a globally defined vector potential to the magnetic field through B~ = ~ A~. However it is ∇ × ~ possible to construct a well defined vector potential in the northern hemisfere (AN ) ~ ~ and in the southern hemisfere (AS ). This is physically consistent if the fields AN ~ and AS differ by a gauge transformation on the overlapping region (the plane of the equator)

~ ~ ~ AN = AS + χ eq eq ∇ 1 Strikingly the Dirac quantization condition can be already derived in the context of classical electromagnetism. 200 Appendix B. Electric-magnetic duality

and analogously for the wave function

i q χ ψ = e− ℏ ψ . N |eq S |eq By means of Stokes’ theorem one obtains

~ ~ ~ ~ ~ g = d~σ B = dℓ AN dℓ AS = χ(2π) χ(0) . 3 · · − · − ZS Zeq Zeq But the wave function ψ must be single valued along the equator requiring from (B.4) χ(2π) χ(0) = 2πℏ n which leads to the Dirac quantization condition, qg = − q 2πℏn, with n Z. The relation (B.3) is not invariant under the continuous duality ∈ transformation (B.2), but it can be generalized to the so called Dirac-Schwinger- Zwanziger condition [57] that is invariant. The latter applies to a system containing two dyonic states (i.e. states possessing both electric and magnetic charge) of charges

(qi,gi) and (qj,gj) and reads

q g q g =2πℏn . i j − j i ij B.2 Non Abelian case: the ’t Hooft–Polyakov monopole

As discussed in the preceding section magnetic monopoles must be included at hand in a purely electromagnetic model and there exist no experimental evidence justifying this construction. In non Abelian gauge theories on the contrary monopoles and dyons emerge naturally in association with particular field configurations. Indeed non Abelian gauge theories admit solitonic solutions of the classical equations of motion that can be interpreted as describing monopole (and in general dyon) states in the quantum theory. The case of the ’t Hooft–Polyakov monopole [180, 181] in the Georgi–Glashow model with gauge group SU(2) will be considered here. The results can be generalized to a larger gauge group G by, for example, embedding a SU(2) factor in G. The Georgi–Glashow model is defined by the Lagrangian 1 1 = F a F aµν + DµΦaD Φ V (Φ) , (B.5) L −4 µν 2 µ a − where F a = ∂ Aa ∂ Aa +eεabcA A is the non Abelian field strength and the scalar µν µ ν − ν µ bµ cν fields Φa are in the adjoint representation of the gauge group SU(2) with covariant B.2 ’t Hooft–Polyakov monopole 201

2 a a abc derivative DµΦ = ∂µΦ + eε AbµΦc. The potential V (Φ) is λ V (Φ) = ΦaΦ v2 2 , 4 a − so that the classical equations of motion are

aµν abc ν DµF = ε ΦbD Φc (DµD Φ)a = λΦa(ΦbΦ v2) . (B.6) µ − b − Furthermore F satisfies the Bianchi identity D ( F )µν = 0. µν µ ∗ The total energy for a given field configuration is

1 E = d3x Θ = d3x (Ba)2 +(Ea)2 + (Πa)2 + [(D Φ)a]2 + V (Φ) , (B.7) 00 2 i i i Z Z a a a a where Π is the momentum conjugate to Φ and Ei and Bi are the ‘non Abelian’ electric and magnetic ﬁelds

F a = Ea , F a = ε Bak . 0i i ij − ijk The energy density Θ is semi-positive definite, Θ 0, with equality if and only 00 00 ≥ a a if Fµν = DµΦ = V (Φ) = 0, implying that the vacuum of the theory corresponds to a a a 2 configurations with vanishing Aµ and constant Φ satisfying Φ Φa = v . The set of configurations corresponding to the Higgs vacuum (moduli space) will be denoted by

MH = Φ : D Φa =0 , V (Φ) = 0 MH { µ } In the case at hand is the sphere S2 parametrized by the ‘coordinates’ Φa MH a 2 constrained by Φ Φa = v . For v = 0 the gauge symmetry is broken to U(1) and the Higgs mechanism 6 produces a perturbative spectrum which consists of a massless ‘photon’, massive √ bosons W ± with mass mW = ev and a massive Higgs scalar with mass mH = 2λv. In addition to the constant vacuum solution the classical equations of motion admit solitonic solutions corresponding to static field configurations with finite (and non-vanishing) energy. In order to have finite energy the fields Φa must approach (sufficiently rapidly) values in for large values of the radial coordinate r, so that MH 2 2 they define a map Φ : S H . For the Georgi–Glashow model one has H = S ∞ →M M 2From now on natural units will be used so that ℏ = c = 1. 202 Appendix B. Electric-magnetic duality

and the corresponding solitonic conﬁgurations are characterized by an integer winding number n Π (S2) Z. It can be shown [182] that the most general solution for ∈ 2 ≡ the gauge ﬁeld is 1 1 Aa = εabcΦ ∂ Φ + ΦaK , µ v2e b µ c v µ

with an arbitrary Kµ. The resulting ﬁeld strength is 1 F a = ΦaF , µν v µν

where Fµν is an ‘Abelian’ ﬁeld strength 1 F = ε Φa∂ Φb∂ Φc + ∂ K ∂ K , µν v3e abc µ ν µ ν − ν µ

1 jk satisfying the Maxwell equations. The corresponding magnetic ﬁeld Bi = 2 εijkF allows to deﬁne a magnetic charge obeying the Dirac quantization condition

~ 1 2 i abc j k 4πn g = d~σ B = 3 d σ εijkε Φa∂ Φb∂ Φc = , (B.8) 2 · 2ev 2 e ZS∞ ZS∞ where n is the winding number. a a The configuration (Aµ, Φ ) is interpreted as describing a magnetic monopole. In fact the energy density decreases to zero outside a finite region as is expected for a ‘localized’ object. Furthermore computing the non Abelian magnetic field 1 1 1 Ba = ε F ajk = Φaε εbcdΦ ∂j Φ ∂kΦ = ΦaB i 2 ijk 2v4e ijk b c d v i allows to associate a magnetic charge to the configuration using the fact that for large a a 2 2 r Φ must approach values in H , i.e. Φ Φa = v on S , so that M ∞ 1 a ~ 1 a ~ ~ 4πn g = d~σ Φ Ba = 2 d~σ ΦaΦ B = d~σ B = . (B.9) 2 · v v 2 · 2 · e ZS∞ ZS∞ ZS∞ Equation (B.9) shows that the “magnetic particle” satisfies the Dirac quantization condition, eg =4πn. Notice that the non Abelian field strength is aligned along the Higgs field outside the core of the monopole. The magnetic charge g can be written in the form

1 a 1 3 a a g = d~σ B~ Φa = d x B~ (D~ Φ) , v 2 · v ZS∞ Z i which follows from Stokes’ theorem and the Bianchi identity, which implies DiB = 0. B.2 ’t Hooft–Polyakov monopole 203

For a static configuration with vanishing electric field one then finds that the ‘mass’ given by (B.7) satisfies the Bogomol’nyi bound

m v g . (B.10) M ≥ | | In (B.10) the equality holds if and only if V (Φ) = 0 identically (i.e. λ = 0 in the Georgi–Glashow model) and the ﬁrst order Bogomol’nyi equation

a a Bi = DiΦ (B.11) is satisfied. An explicit solution for the monopole field configuration can be found in the special case of spherical symmetry and if the Bogomol’nyi bound is saturated. A field configuration corresponding to a non trivial vacuum cannot be invariant either under the group of spatial rotations SO(3)R or under the SU(2)G group of global gauge transformations 3. However a solution invariant under the diagonal SO(3) subgroup of SO(3) SU(2) can be obtained through the ansatz R× G xa xj Φa = H(ξ) , Aa =0 , Aa = εa [1 K(ξ)] , er2 0 i − ij er2 − where r = x , ξ = ver and the functions H and K satisfy K(ξ) 1, H(ξ) 0 for | | → → r 0 and K(ξ) 0, H(ξ)/ξ 1 for r . In the limit in which the Bogomol’nyi → → → →∞ bound is saturated an explicit solution was derived by Prasad and Sommerfield [183]. The exact form of the solution will not be discussed here since it will not be relevant for future considerations. In general a non trivial vacuum in the limit V (Φ) = 0 (BPS limit 4) is rather un- natural and moreover the condition of vanishing potential is expected to be broken by quantum effects. However in supersymmetric theories this situation is perfectly consistent because of the presence of flat directions that are protected by supersymmetry. The construction leading to the monopoles considered so far can be generalized to the case of non vanishing electric field giving rise to configurations with both electric and magnetic charge (dyons) [184] 5; for such states the Bogomol’nyi bound becomes

m v e2 + g2 . D ≥ 3 Gauge transformations that do not reduce top the identity at spatial infinity are referred to as “global” or “large” gauge transformations. 4Field configurations saturating the Bogomol’nyi bound are called “BPS saturated”. 5Static dyon solutions corresponding to finite energy configurations cannot be constructed in the temporal gauge A0 =0 204 Appendix B. Electric-magnetic duality

The Lagrangian (B.5) can be modified by the introduction of a θ-term θe2 = F a F µν (B.12) Lθ −32π2 µν ∗ a without spoiling renormalizability. The addition of amounts to consider a com- Lθ plexified coupling constant θ 4iπ τ = + (B.13) 2π e2 and write the Lagrangian as 1 1 = Im τ (F µν + i F µν)(F + i F ) DµΦD Φ . (B.14) L −32π ∗ µν ∗ µν − 2 µ In the presence of a θ term the dyons acquire an additional electric charge [185] (Witten effect). More precisely the electric and magnetic charge operators are 1 Q = d3x D ΦaBi m v i a Z 1 Q = d3x D ΦaEi (B.15) e v i a Z and one finds for the electric charge of a dyon eθn q = n e m , e − 2π eg with n Z and n = Z. e ∈ m 4π ∈ B.3 Monopoles and fermions

In realistic models the gauge fields are coupled to fermions and one is then led to consider monopole configurations in the presence of fermions. The coupling to spinors in a monopole background has important consequences that will be briefly discussed here in the simplest case of gauge group SU(2). Dirac fermions can be coupled to the Georgi–Glashow model by adding to (B.5) the term

Nf k k = iψ γµD ψk ifψ Φψk . (B.16) LF µ − Xk=1 The Dirac equation reads

(iγµD Φ)ψk =0 µ − B.3 Monopoles and fermions 205

and a suitable choice for the γ-matrices is

0 i iσi 0 γ0 = , γi = . i −0 −0 iσi In the background of a monopole one can look for a solution for ψk in the form

ψk(x, t)= eiEtψk(x) ,

k χ+(x) with ψk(x)= . The Weyl spinors χk then satisfy the equations χk (x) ! ± − k i k k Dχ/ + = (iσ Di + Φ)χ = Eχ+ − k i k k Dχ/ = (iσ Di Φ)χ+ = Eχ , − − − which in general have both positive energy (E >0) and zero energy (E=0) solutions. The monopole ground state is then constructed as a combination of the zero-energy eigenfunctions that are interpreted as ‘fermionic collective coordinates’ describing a Grassmannian deformation of the original monopole. The generic solution of the Dirac equation can be written in the form

n ∞ k i k j k ψ = a0 ψ0 i + a ψj , (B.17) i=1 j=1 X X k where ψ0 i are the zero-energy eigenfunctions. The coeﬃcients of the zero-energy eigenfunctions in the expansion (B.17) become creation and annihilation operators and zero-energy solitonic states are obtained by acting with such operators on the “bosonic” monopole ground state. The number of fermionic zero-modes is determined by an index theorem [71]. In the case of fermions in the fundamental representation it can be proved that there is one zero-mode for each ﬂavor for a charge one monopole. Rewriting the Nf

Dirac fermions in terms of 2Nf Weyl fermions one ﬁnds that the creation and annihilation operators generate a Cliﬀord algebra of dimension 22Nf /2 = 2Nf , that splits N 1 into two irreducible representations of dimension 2 f − . As a result the monopole ground state is a spinor of SO(2Nf ). The angular momentum generator for a symmetric monopole is K~ = L~ + S~ + T~, where T~ are the SU(2)G generators of global gauge transformations. Fermions in the fundamental can have ‘spin’ zero since they transform under SU(2) SO(3) in the representation 2 2 = 1 + 3 which contains G× R × 206 Appendix B. Electric-magnetic duality

the singlet. Hence the fermionic zero modes carry zero total angular momentum and the resulting monopole configurations have spin zero. If the fermions are in the adjoint representation there are two zero-modes for each flavor in the one-monopole background [71]. Furthermore these zero modes necessarily have K~ = 0 since the adjoint fermions transform in the 3 2 = 2 + 4 6 × of SU(2) SO(3) . Since the zero modes are two-fold degenerate they must carry G× R spin 1 . Consequently in the simplest case of N =1 the corresponding creation and ± 2 f annihilation operators (that will be denoted by a† and a ) can be used to generate ± ± the following multiplet of zero-energy states acting on the charge-one Clifford vacuum Ω | i

State Spin Ω 0 | i 1 a† Ω +| i 2 1 † a Ω 2 −| i − a+† a† Ω 0 −| i

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